Inverse Laplace Transform (ultimate study guide)

yes we all know last time I did that lap last marathon and he wasn't be fun but have you guys ever wonder shouldn't we be doing the inverse Laplace inverse as well and of course the answer to that is yes so in this video here is the impersonal plus marathon and we have a total of 24 then you know you're honoring copyrights of total or 24 and this fire is in the description for you guys already you could download that and work your own way right so let's just go ahead and get started right here this is the notation for the inverse Laplace and you see that the inside isn't pressure in terms of the s right so we have the function vs world we have to go back to the G world now before we start actually I have to tell you guys here's a challenge when you're doing the Laplace for the inverse Star Plus especially for inverse Laplace you have to know how to write the s and also the v really well by can't so here is the deal alright in this video when you see this this right here means 5 and if you see this this right here is the s a seeing super or supreme or whichever water you like with s right so that's my challenging as well in this video all the assets will be in red right so you can see sounds like this I'm trying to say that suffice anyway enough talking going back to the S word just remember the same stuff with you really well and we just have to go backwards this is the one that you have to know from what we did last time note if you have the original Laplace of T to the N power this right here is equal to n factorial over s to the n plus 1 power like this right I get just how to remember the formulas and all that stuff and you see this raise your mattress that when we have to go backward we can expect you have T to s power or so now look at this and look at that we can see the end has to be 3 but the promise that we do not have 3 factorial on the top but it's ok this is how we can do so here we go they meet this register again here we have the interest are PLAs on the bottom we have s and then again for its SMS three plus one that's good and is equal to three no big deal but originally when you have a one that's not good but it's okay because we just do the following here is the one that we have but we can actually just multiply by 3 factorial to make this into that form right however if you just multiply by 3 factorial of course this is no longer the same as the original but it's okay because Laplace is linear all we have to do this on the outside just go ahead and divide it by that comes the motor pole I just multiplied it then we go ahead and divide it by 3 factorial so that's pretty much it and as you can see here worked out 104 3 factorial 3 factorial is 6 because we times 2 times 1 so we have 1 over 6 and then for all this and yes again 3 so going back we will get T and the end is equal to 3 so we have 3 the third power right here and that's it right so this is the one that you have to remember and later on we'll just cannot do this in our head now question for you guys do we need to put on plus C no good all right this is the infrastop class all right anyway first one pretty cool huh now second one yeah we have the inverse Laplace and we have 1 over 6s plus three 6s plus 3 all right in my opinion really II thought if you assign a question that 104 5s plus 55 what's nothing like that see you stop complete ok so for this one hmm what use a different one well here is the deal our just recall all the things that we did from the previous video for you guys the original inverse sorry the original Laplace of e to the 80 power so again I just made blue 80 power this right here will give you one over s minus whatever this a is so we have one over s minus a but here is the trouble right here we have a six times s so to fix that likewise this is the plus this is a minus so we also to fix this but it's okay we can just do algebra for this so here we go well we have the sixth right here and again just a hole we did earlier we can multiply things and devices and all that and in this case we can factor out the six and then we can take that outside of the infrastop class because laplacian first up class they are both linear but stuff so let me just write this down right here this is the original question the infrastop plus one is the on the top over this six i had to factor out to the front so let me just put down as 1 over 6 and then 6 s earlier now we will just be s right here but here it was a 3 but with right other state so it becomes 1/2 so remember that I so technical use of parentheses like this if you want to make it super clear and then the plus we'll look at the plus s minus minus like this so we have a minus minus one over two like so and that's pretty much it this right here is the phone that you need to go back to that you world well here we have the 1 over 6 and to go back of course it's just e to this power times T therefore we have negative 1/2 right here and then we multiplied by T like this that do we need a policy of course no this is the impersonal plus right so that's pretty much it right so just continue with number three I think I should be able to write down another one yeah let's do another one right here number three we have the following we have two infrastop class of s plus 1 again as in red plus one over s in red and then plus two like this right now here is the deal on the pattern you can see that we have a square plus two this is the time that were to use a sine or cosine and then we had to use both you will see why here let me just write on uh no again for you guys okay remember that the original applause of sine of BT right this right here gives you B over s squared plus B squared like this and then we also have the Laplace of cosine BT this right here gives you s over s squared plus B squared like that so these are the two that we have used well I'm not happy we have s plus one on the pattern we have this right so we'll have to actually break down the fractions so here we go for the first one we already done the inverse Laplace and then let me just write s over that so we have interest applause and then it says s over this is s square plus 2 well because of the formula like this we have to really look at this as B Square so don't look at the two as to look at the 2's square root of 2 squared so this right here I will add square root of 2 squared so that's actually the first part right here right now we add again inverse Laplace is linear so that means you can break this apart so the second part is the inverse Laplace and the denominator will stay the same so we have this again s square plus will right here square root of 2 squared as well yeah and right here on the top we have 1 so that's what we have right this is just the same as this however in order to go back to the key world the first one is ready but the second one this right here is ready why because the table right here we should also have to be when we go back to the sign of B team but it's okay because we can do the things just like this right here we can multiply and just remember that we have to divide that's all so here is the deal this one is ready the B is just square root of two so that's very nice however for this one this is square root of two on the top I really want to see the be right here as well so I will have to multiply by square root of two and then just remember to divide it by square root 2 in front as well so that's all we have now we can go back nicely the first one is the cosine one with the S on the times the cosine one so we have cosine and by the way when you have the cosine yes that's how to be in red all right life doesn't have to be the heart closed sign and then the B is square root of 2 and then of course you have to have the T and then here we have the plus 1 over square root of 2 and then this is the same version so it's just right on sine of square root of 2t and hopefully one day I know how to hold two markers in one head and I'll be so cool by the way I don't know here is the answer for the third one like this right so pretty cool huh and again this should be a clothespin to see you guys can see that all right so this is pretty nice this is the infrastop class transport and I don't know if you prefer this over in the gross whatnot but unfortunately for the infrastop last we can only do so many in terms of how many types and by the end of this video I'll show you guys how to use the infrastop plus and also the Laplace transform to solve differential equations so you haven't done that before this is the time all right so the meter is right here yeah so here is number four here we have in first applause of this and that and again let me know how you guys are doing hopefully everybody is doing well all right yeah we have one over and this is s square plus twist s square plus 2 s now let me just tell you guys when we're doing the inverse Laplace transforms you it you are buddy to know you are partial fractions really really well and of course we have done that a lot by in the calculate and those think equal days and really need to know how to use a car parameter it's going to save you a lot of time so just like this one s square plus 2 s we can factor the s on the pattern first right so if you do that you see here we have s and then we will just get S Plus 2 like this so this is a product two distinct linear factors right here we can just register in first applause and let me just put down like this to do the partial fraction we all know we will have a number over the first factor which is just s plus another number over s plus 2 like this right and now here's the partial fraction with the top right method to figure it out this number which has the S on the bottom we go back to the original and cover up the same denominator and you ask yourself how can you make s equal to zero we're just told you the answer as has to be zero and then you plug in 0 into this right here and I just worked out 1 over 300 passed through just one half so we just get 1 over 2 on the top this is the 1 over to the top right and then to figure out this number you would go back to the origin and cover the same set in a meter and you answer us up how can you make s plus 2 equal to 0 s has to be negative 2 and you put it right here and then we will have 1 over negative 2 so this number is just going to be negative 1 over 2 like this right and then of course you can just go ahead do the following well here's the deal this is s to the first power right which is the same as saying s to the 0 plus 1 so it's going to be T to the 0th power on the top is technically you can't just do Cadiz s 0 factorial well anyway whenever you have 1 over s to the first power when you go back to the few world it's going to be a constant so that constant in this case it's just 1/2 times 1 so just 1/2 right so again if you take the original a part of this you will get back to that right next let's put on the constant multiple which is the minus 1/2 and this time this s plus 2 is the same as saying s minus negative 2 right and this is the one that has the e right e to the a T power and the a is negative 2 so we have e to the negative 2t like this so again know your partial fractions really well this is that the combination of question 1 in question - right ok now let's do care the tip 1 we have the first term plus oh ok this is 5 oh we have s over s plus 2 over s plus 2 square like that right hmm we can do some partial fractions if you like or you can do the partial fractions this way in math the most important thing is that you need to know how to make things happen sometimes it just worked out so nicely if you can just see the picture and just kind of walk you out yeah show you how here is the deal let me write this down again right so let me just show you here we have 2 inverse top-class let me just write down all original again so we have s over s plus 2 squared yeah now this is what Y will this is what we'll do on the bottom this is s plus 2 squared so that means s plus 2 s plus 2 yeah and just thing about it ask yourself this question all the time right wouldn't be nice if on the top we have s plus 2 as well yes why because as class two and one of the s plus 2 can cancel out right that's really good well in that case can I produce the s plus 2 like so sure I can right why not I can just add 2 to it but that changed the whole thing that's not good but don't worry as long as you remember to minus 2 it's okay right because they are equivalent now check this out again we will break this apart look at this over that this and our cancel so we actually just have the following in first our class and let me break this down for you guys we have one over just this to the first power right just this to the first power because again this and I can saw which is very nice then for this power here we have the minus two over that so the mean just put a minus two in the front and then I will write down the infrastop class again and then we have the one on the top over s plus 2 sorry s plus 2 square like this right so that's what we have now take a look right here and this is the one better so Tampa what's what the answer is let's go ahead and just think about and it just you know say it what's up let's the enter for the first one we cannot do that over there already right okay so here's the deal okay this is just as plus two to the first power so at only that this is the same s as minus negative 2 on the bottom which is what we did earlier just that part so we acted out with e to the negative 2t yes now for this is studied trickier so pay close attention to this we have the minus 2 right here that's good now we have 1 over s plus 2 squared hmm can we don't worry I got you here some notes for you guys not this time I will have to write this down for you guys right here if you have the original Laplace notice we have s to some power yeah so you can expect to use T to some power original applause of T to the N this right here is n factorial over parenthesis well over s to the n plus 1 power right good now here is the thing if you look here this is brushing the s world we have AZ plus 2 inside so this has been shipped so I'm adding a 2 right here and this is what you have to do right here if we multiply by e to the a T then what's going to happen is that you just to the shipment in the S world so Luke awadhi s is right here and then just go ahead - that a value and then you raise that to the n plus 1 power like that so that's one of the properties so you actually get this right here s minus a and then raised to the n plus 1 and then all the stuff so as you can see in this case our n is just nice to equal to 1 yeah so that's good on the top we have one already so that's wonderful so we will have tea to the first power right here key to the first power again TC n is equal to one but then we have the a is negative two so just remember to multiply by e to the negative two and usually we put down the T first so that's why I'd write it down like that so T to the first and all that and we're time so I can start that's it right so number five right here all right so question for you guys do you guys remember all your in first do you guys remember all your Laplace transform table honestly I don't because I I do a lot but like I don't because they are some of the Laplace transforms I think that happened time yet so those are the ones I don't know of course but the ones I did in the previous video I do remember that all right so number six I only have the answer key just to make sure that I can provide the answer key to you guys as we want to make sure my answer keys it's correct so that's why I'm so happy I think you are here in front of me anyway inverse Laplace of s oh man this is the crazy one s e to the negative PI over 2 s over s square plus one okay so here is the deal have a look this is e to the negative PI over 2 s we are talking about in the s world the one that with the earlier was in the T world and I think it's a good idea let me just remind you guys the formulas again right so here they mean just put this down for you guys one more time the one that we use earlier is when you have the original class if you have e to the a power times T to the nth power look at this puffers which is n factorial over s to the n plus one yeah and then because we have e to the 80 right here this tells us we had to subtract a and then raised to the n plus one that's the one that we used earlier however though this right here is really confusing this right here it's in the s world so the just thing about the original Laplace of what function will keep you in to the some number times s in the s world into the the me just read yes so the e to the whatever parts in the wrong world right so what is this you have two answers for this the first answer is this right here you can have the Delta of t minus a Dirac Delta function without a team yeah or another one that you have to keep in mind if you have the Laplace transform of the unit step function of t minus a this right here will keep you this part which is e to the a s but then you have to divide by s oops sorry to mistakes of RDS as being read right so these are the fundamental ones and just remember that however though this right here is not quite the same as that so in fact hmm what do get the harder version of all this one more for you guys when you have this right here this is the time that you usually end up with e to the a s power this right here if you multiply by some function I'll just put on F and then if they have the same input P minus a like this then what you do is this part is the same as that which is e to the a s but you look at this part and just multiply by the Laplace transform of F of T right if the inputs right here they match this one here just F of T like that and this right here is the general case this right here you just take F to be one and it's a plus of 1 is just 1 over s right all right so this is the one that we had to focus and now this is how you can do it whenever you see e to the number times s in the s world yeah that's what we have right here this is Sonya start just go ahead multiplied by U which is the unit step function of t minus whatever that number is actually Billy kappa is this a and that a match right why do you see it just well okay just - t - no sorry sorry so this right here is meant to be negative is negative is I believe negative a is let me see let me see yes because if this e to the a s that wouldn't even converge so it should be negative and that's what be negative and that should also be negative right so anyway go right here this number just go ahead and subtract whenever you see e to the negative pi s right here just go ahead and subtract the PI over 2 PI over 2 just like that all right so this again let me just recall the formulas Delta you can e to the negative a s if it's a unit step function you get e to the negative a s over s and in general if you have a function and the unit step function you have two inputs are the same you go ahead and do this e to the negative s times the Laplace of the function f of T right and this is the one that were using whenever we see e to a number times s in the s world and notice this is negative a so the area is PI over 2 so we are just going to multiply unit step function of t minus PI over 2 like that all right we are going backwards and that took care of the first part now we also have to figure out the rest this is for that now if you do focus on the rest I will just put this down here for you guys so note just note how great on another note for you guys in this video now to look at the remaining part which is the inverse dot plus the remaining part is just s over s square plus 1 and this right here it's very nice because this is nicely equal to f of this is nicely equal to cosine of T right this is just cosine T well this right here if you look at the right-hand side remember we're taking the inverse were looking right hand side and then do kidding Sarah here so in here this is actually our f of T ok however when we go back to the T world and put on the answers you had to have f of t minus a as well the input have to match right here so instead of just put down cosine of T you actually have to put up cosine of t minus PI over two right so again this portion is for the F of t minus a so I'll put this down here this is t for t minus a and this is the unit step function of t minus a so again what doing backwards like this alright so this right here is the sixth one like that I'm sorry earlier I forgot a negative right yeah all right so that's number six and then now let's see number of seven yeah I'm going to this alright number seven in action here we have the impersonal plus and the input is s over s squared plus 2 s plus 2 right so that's number seven recording good now on the bottom here s square plus 2 s plus 2 ask yourself can we factor it the answer to that is no unfortunately not the worst part is that the pattern right here doesn't even have real roots so what we have to do is complete the square so let me do it on the side for you guys so right here CDEs of this expression which we have yes where oops this is meant to be in black square so s square plus 2 s plus 2 right so what you do is you leave it right here and then just add 2 after that right and then what you have to do is the following make sure this is the one in front of the s square which it is and then look at this number this number is 2 so what you have to do is take half of this number which is 2 putting in parentheses and square that and work that out which is just equal to 1 so this is the magic number that you have to add and also subtract all right so this is the general way to do it I know some of you guys might be able to just see it and write down the answer that's fine of course anyway after you have done this here you will end up with the first three terms is as class 1 and then square and then this is just going to be plus 1 so that's what we have so that's the first thing here we have the Inc first applause on the top we still have yes so go ahead and write that down over on the bottom we have this which is s plus 1 square and then plus 1 which is the same as 1 square because you know sine and cosine are about to come out so this is what we have good now if this was just s square plus 1 that would be cosine right but this right here is s plus 1 so do it really really carefully and again write down some notes right here for you guys here are some of the things like that at all again if this is just s then you have the following right it's just cosine because we know the original Laplace of cosine T this is nicely equal to s over s square plus 1 because the B value is right here is just equal to 1 so that's what we have however as you can see we are shifting in the S world this is s plus 1 so you have to remember is we multiplied by eat if you eat something the cosine if you multiply the cosine T with e to the 80 then what you do is look at all the s in vs world and then you go ahead and shift it and you do as minus 8 here and as minus a here so that's what you have right similarly for sine s will but here is the thing this right here match with that is negative 1 on the top I don't have the plus 1 right this is s plus 1 need to match it so that I can do this right so what do what we did earlier right it's done okay so here we have the interest of our on the top is 2 s on the bottom let me just write es s plus 1 square plus 1 square now I really want to have the top in this match so go ahead just add 1 so that's how that you can in Comanche listener match now again as usual this will change the whole thing that's no good it's okay just go ahead and minus one so as you can see if you look at the interest of this portion which is s plus one if you just look at s plus 1 over s plus 1 square plus 1 square cause that this is what we can do and then we can just go ahead and just minus I this is minus and then we have the infrastructure pretty good now because we can actually write down the answer for the first one again you see we have s minus a so this is as plus a a has to be naked he want so what you do is go ahead and write down e to the negative 1 t that's a shift team and then the form is the cosine so that's the first one next - well as you can see on the top this is just one there's no s so this is the same person right and again for this part right here it doesn't really matter because the sign on the top is just a constant multiple just a 1 right and this is one square so this and that match already you know you end up with sine T and all you have to do to make the shift in the s world is multiply by e to the negative T so this right here yes get you to the negative he order stuff so that's it now next one take some space so well so this is a great review if you did a Laplace transform ready of course you should also know the rest of us now number eight right here this is number eight right number eight we have the Laplace and this is inverse of course one over s to the third power and then s square plus 1 like that you have two ways to do it first what you can do actually you have to do ways to do it first way you can do partial fractions I'll leave that to you guys because in order to have 1 over s plus third power right here when you break down right partial fractions you will have to do a over s to the first power plus B over s to the second plus C over X to the third power you have to put on a power and then you add this is an angry to support quadratic so you have to do DX plus e over s squared plus 1 again other work you are fine ounce ABCDE I don't think I want to do that so let me not do the partial fractions second one you can use the convolution theorem and Interpol for that might be manageable might be okay but I actually have a convolution idea for you guys later so let me just give you guys a quick review right but let's see I would just tell you you can do partial fractions let me just write down the things that we have partial fractions and again if you do the partial fractions the breakdown for this is that you have to first get a which I don't know all 4 s plus B over s square plus C over s third power the s+ over s square plus 1 let's not do that alright so that's the first way first second you can use the convolution tongue for lucia 0 and what is is the following notice that we actually have a proton inside namely 1 over s to a third power times 1 over s square plus 1 and the convolution theorem says the original version is the original Laplace of F of T star this is the convolution this right here with G of T this is going to give you very nice F of s times this is just a regular multiplication times G of s oops sorry yes here and that's here so that's good and of course you can do it backwards so to do that backwards you are to the following and by the way let me remind you guys the convolution is an integral so this right here what you do is integral from 0 to T and then you end up with F of T minus V G of V DV anyway if you don't do the convolution for that you will look like this that's right on the infrastop class look at this as 1 over s to the third power times 1 over s square plus 1 right well what you can do is you can do the first one now you can do the first one you can break the apart but inside the regular multiplication and outside here it's going to be the convolution because we are going backwards and then you have the inverse Laplace of 1 over s squared plus 1 and now let's do this in our head this n is equal to 2 so that means that you have a 2 factorial not half by Doh so I'm going to multiply by 2 factorial and then go ahead and divide it by 2 factorial yeah good so this is actually just 1/2 so that's nice and then this is T Square that's good and then you have to do the composition so again you have to come pollute it I don't know if that's a word and then he took this right here which is going to be sign and then you end up with the integral which is actually zero to T this is the F so you actually have to put t minus P into this right here so you actually have to do one half t minus V Square and then regular multiplication inside and then G of E which is sang fee the fee and just have to work that out so yeah you have to work that out unlike hm start do not just put on three dots you actually have to work that out and show the teacher how the answer is right but this is how you can approach that I have a better example what's the competition later on so I'm not going to do the compilation not not compilation convolution this is what we can do here is a third way we're going to use the following okay hmm well we are motor what / s huh so I think there's a pump law that we have seen that's dividing s on the outside that's actually this one right here if you look at the integral and if the inside it's an integral which is the integral going from zero to T and as there we have F of T TV this right here if the inside integral then the outside will give you F of s F of s divided by s like that in other word if you take the inverse of this expression then you just have to do the integral of that so you see we actually just divide as two times and this integral is study less intimidating than the convolution integral in fact this is actually the repeating integral formula I have a video on that if you want to go check it out but not to be strong to my daddy let me just show you guys how we can do that right so here is step one right so this is what we have to do we know the Laplace inverse of one over s square plus 1 this is nice to equal two sine T so this is just for we know right then what you can do is the following what you can do is the following then if you want to do the inverse Laplace of 1 over s squared plus 1 yeah same one but if you really want to have an S right here is multiplied with stuff which is the same as you're just dividing by the S right we're just dividing by this addition of s so this has chips them just write it down great if you really want to divide it by as right here as well like that what you are going to do is this in this right here one of us you can just go ahead do K what this is and then integrate from 0 to T and change the input to be V so we have sine V DV like that so sometimes it might be a good idea to integrate this little ones more times than to a crazier one but this is actually not so open either because you can just induce integration by parts in fact that's the connection between that repeating integral formula process the forces the this way okay so you have to integrate these three types all right so what I'm max just cubic Mara and because I have to erase this now and I just have to integrate these three times now so bad or maybe you like this more well maybe you like the convolution I don't know right now let's go ahead and do that again one of us is you do the integral so just go ahead and do the integral in the cosine is negative cosine so we have negative cosine V and then we are going from 0 to T so be recap I'll put him here first we get negative cosine T and then put this right here we can - and this is negative cosine of 0 so in another word this right here is equal to negative cosine t minus 1 not plus 1 so that's the first one so I'm time showing you guess this one I just stopped in for surplus you don't have to do the partial fractions if you two want to anyway you will see that the next one the in first are plus if you have the 1 over if you have the s times s square plus 1 already which you know if this right which is just equal to negative cosine T plus 1 I'm using T because I will have to integrate that well right here if you have 2 divided by as again just go ahead integrate this again from 0 to T and then you have this in the world and just pretty much go into that so let's go ahead do that integral of negative cosine it's going to be negative sine V integrating this is just plus V and then we have to plug in plug in 0 to T so plug in T we get negative sine T plus T plug in 0 plug in 0 is just mine zero right so that's pretty much the answer right here she writes the result it's just yourself so that's the answer so the last one here it's going to be the universe well here is one over s square times s square plus one yep and then again to give us their power of course we have to have the 1 over s right here so this right here this portion just give you that which is negative sine V Plus V and we have this so just go ahead and integrate from 0 to T and then in the v-world like that all right then just go ahead and integrate this the integral negative sign fee it's going to be positive cosine P and then integrating down we get plus 1/2 V squared and then again plugin plugin so plug in 0 and T so in this case our actually is ready down here for you guys this is going to be the following plugging keep locking P so first we K cosine T plus 1/2 T squared this is the first portion and then minus the second portion we have cosine 0 plus 1/2 0 squared like that yeah so finally you see here is cosine T plus 1/2 T Square and this is minus 1 right and that's zero so finally the answer is this cosine T plus 1/2 T Square minus one like that yeah and of course you see one half he's queer yeah so just like this it seems that make a typo mantequilla will have to because I have encircled right here right so one half he's queer I think I actually had an extra cosine T here which I don't think is correct on mind so let me double check because maybe I distance is not Europe I don't think so yeah t-squares never multiply with the cosine so I think this right here minus 1 yeah yeah all right so that was number 8 all right now moving on to number nine number nine right here here we have the inverse Laplace and we have one over I prepared the S Plus - hey I messed up because I didn't want to choose Phi Phi this is my Phi right so hopefully this is clear so this is s that's five finally this one right here so this right here is different than the one that we did earlier you don't do partial fraction at all because this is just a shifting situation all right so let's practice this one more time just pay attention to the input just pretend this is s to the fifth power in that case what we need is the following so again note if we have the Laplace of T to the nth power this will be n factorial over s to the n plus 1 power s to the n plus 1 power so this is the one that would keep in mind however it's being shipped right so again you have to multiply this by e to the a a t and that will be as - a raised to that power so that's right here so now right here let's just finish this right here we can see and has to be four but I don't have full factorial on the top it's okay let's just go ahead and multiply by 4 factorial and just don't forget to divided by 4 factorial all the way on the outside now let's write that down 1 over 4 factorial that's the same as 1 over 24 and then for this part of course we know we will just go back so that's Z to the fourth power that's write that down first T to the fourth power and then a yes there could be 2 so just put over here so we have this and then e to the negative 2t like that ok so sometimes you do partial fractions sometimes you just have to remember this form what this is but pores just kind of yeah this is the ones that should not be forgotten yeah all right number ten though this is the thumb one we have the inverse Laplace of one over square roots haha square root of s plus 1 over square root of e to the S yes how can we do this well this right here it's the same as Laplace of T to some power right so they mean just remind you guys this right here again and this is not shifting so I don't have to worry about the ETA T part this is just going to be the following so note the original Laplace of T to the nth power this right here is equal to n factorial over and we have s to the plus one right so in this case you can just go ahead and write this down here we have the inverse Laplace for the first part we can look it as 1 over s to the 1/2 power so we can use this to take care of the first one yeah hmm and plus 1 has to be 1/2 so n is equal to 1 plus 1 is equal to 1/2 so n is actually negative 1/2 right but on the top I do not have negative 1/2 factorial it's ok let's just go ahead and do the same thing let's go ahead multiply the top by negative 1/2 factorial and then on the outside just remember to divide it by negative 1/2 factorial right and you might be even wondering that what in the world is negative 1/2 factorial because this formula right here it's only good for 0 1 2 3 4 5 and so on for the end right however if you extend the cost about the N factorial this right here is a same s a gamma of n plus 1 over s to the n plus 1 power like this and if you want to have negative 1 over 2 factorial you can just put that right here into N and you can just put that right here into n so I would just have to tell you right here gamma of let's see let me just read it on like this negative 1/2 factorial huh this right here it says MS you could make it to 1/2 right here which is the gamma of negative 1/2 plus 1 like this and then you can watch my previous video and all that stuff this right here is gamma of parsley 1/2 and you end up with a nice number it's just square root of pi and again what exactly is this though you have to remember this right here it's actually an integral so our actually just remind you guys if you're interested to work that off on scratch this is the integral going from 0 to infinity and you actually have to do t to the 1/2 power which is 1/2 hours right here I T to the 1/2 power and then oh sorry 1/2 power minus 1 because I'm looking at the gamma function so this right here yeah and then e to the negative T and then we have DT like that yeah so if you worked out you actually end up with square root of pi so this is right here it's actually just goodwill PI so is that so just keep dying man pretty crazy stuff I know okay however what's the second one up for the second one this is just the inverse Laplace and we can do some exponent business the square root is the same as 1/2 power e to the S to the 1/2 it's the same as e to the 1/2 s in the denominator we can bring that up so this becomes e to the negative 1/2 yes right again square root in the denominator that's how we can negative 1/2 and then what's this like that well this work is almost done this right here where to remember the Delta function remember when we have the inverse when we have the original let me write on original when we have the original Laplace of Delta t minus a this right here gives you a to the negative yes very very nice right so I just have to go backwards now we all turn them into straight on the answer here is 1 over square root of pi gustavo now here again the n plus the interest radius and is equal to negative 1/2 yeah the Animus negative 1/2 so this is the same as T to the negative 1/2 power done should you just put this down here and was let me have this because we want this to be down right so that's why I wrote down earlier you see could 1/2 so that means n is equal to negative 1/2 so that's how we get 1 negative 1 have right here for this a is 1/2 so it is pleased right here so we just have to add the Delta of t minus 1/2 like this and if you want to be fancy you can put this down showing that so finally you have 1 over square root of pi T and then plus Delta of t minus 1/2 just like that come on yeah so have a look have a look have a look alright that was number 10 and of course we are moving to 11 right here number 11 unless you guys oh no I dropped a cap and I'm not going to pick you up number 11 face ok infrastop plus of s minus s + 8 3 s plus 8 over on the body here we have s square plus 2 s plus 13 hmm let's see always ask yourself can we factor the denominator in this case unfortunately not so let's try to compute a square and that's actually how we have to do it so let's do this in your head right this time this shouldn't be too bad this right here is just s square plus 4 s and this is one that's good and this is for take half of that is 2 squared is 4 so I should have that half for right here now original US 13 4 is 13 so same as for past 9 so this is just like this yeah so of course the first part completing square is done so I write it down here for you guys here we have the interest our plus and on the bottom here we have this on the bottom here we have s plus 2 squared so s is in red s plus 2 square plus nine which is the same as 3 square yep and then that means you're ready down the top which is s plus h y s+ h doesn't change it on the top actually I'm not going to put on plus 8 plus 8 is not nice why because yeah we have s plus 2 so I really want to have it as plus 2 on the top right here is s plus 8 so I'm going to braid yes as plus 2 and then of course we all know 8 is the same as 2 plus 6 so this is how we are going to break down this over the denominator and that over the denominator right so I will read it down one more time for you guys here is the inverse Laplace this over that which is s plus 2 over parenthesis s plus 2 square plus 3 squared very nice and then right here we will add 6 is the constant multiple will put that outside six inverse Laplace and then we just have a one on top and then here we have the S Plus 2 squared plus 3 square like this ok remember this has been shifted originally it would be the cosine situation right but it's being shifted so this is how we do first you see that this right here it's going to be cosine of 3t right imagine this and not just input that get maybe like a big s or so this right here you get cosine of 3t but it's being shifted and then aide right here is negative 2 so you have to remember to multiply by e to the 2 now for this one well this is nuts already yet because on the top here is just 1 I really want to see what as you do want to see a 3 now still factorial is just 3 because this is the sine situation it is P over something square right the function square plus the same number squared so I really want to see the three well multiply it by the street but just remember 2/3 right so you will see that this is actually Plus this and that's two and this portion now is just a signed version so it's time of three T and then we have to do poverty over there right of course so just multiply by e to the negative 2t and we are done I'm not going to ask you guys doing to put on the plasti like 20 something more times oh sure yeah that's it that's it all right okay okay now for the next one it's a crazier partial fraction one so it's a lot of algebra and just have to be patient because don't already to do differential equations you have to have a lot of patience and also if special linear algebra you have to do you have to have tremendous amount of patience all right number 12 right here inverse Laplace of 1 over s to the fourth power fourth power plus I'm very sorry I had to do this this is 5 and then s squared and then plus 4 right and then notice not yes this is just that all right ha we can factor out the bottom that's nice if we do so this is going to be the C treatises you know quadratic you get s square plus 1 times s square plus 4 yeah and it works now again you have the competition way to do it if you prefer to do the integration by parts and just do the integral you can really do the convolution but this one let me show you guys the parts of fractions let me just show you gets a partial fraction for this so partial fractions in action like this way here's partial fraction f doesn't do nice yeah Sally but anyway here we go this is one over and we have s square plus 1 s square plus 1 and we have two protons into two parts so the first one is going to be a square plus one but the thing is that this is the irreducible quadratic on the top we must have a S Plus P tank near linear similarly the second one is an irreducible quadratic so on the top we have have C S Plus T crazy stop now Kling the fraction so multiply everybody by this end that and then let me you know just go ahead and do that just multiply everybody by that so you actually end up with one and that's equal to this times that pretty much so we have a s plus B times s square plus 4 plus C s plus T times s square plus 1 all right and then the deal is that just multiply this out and hope for the best you end though it's a 4.4 maybe with system of equations so unfortunately copper doesn't work because you cannot copper up and then make a term a factor equal to zero and that's not to use complex numbers but don't you could let me know but none now anyway so here multiply this and that we get a Q and the truth is that didn't work to stop before and so up Justin anyway a as cube and then this times that is plus 4ei s and then this x - + B s square and then + 4 B and then let me write down this below so this is gonna be C s cube so we add C s cube Plus this and that which is CS and then Plus this and I ste s square plus lastly we have + T like this right so now let's just combine terms and all that stuff so you can see we have the a + C at the NDS third power + that's right at least our next which is B plus T and then s square item plus 4a + C and then s and then lastly we have the 4 B plus D it's a constant right and of course you can see that we have the 1 1 1 other stuff now what that means is that we must have this right here equal to 0 because the left hand side doesn't have the s to the third power so we must have a plus C equal to 0 likewise we also need to have B plus C equal to 0 likewise we also have to have 4a plus C equal to 0 and then we also need to have 4 T plus T equal to oh this is the constant so we actually have to have it equal to 1 right now if you see it you might be 4 by 4 pi it's actually pretty nice because this and that well if you just look at these two equations they only have the they only have the quote they only have a and C so found here you can see that C is the same as negative a and then you can just do it so you can put this right here so that means for a right for a minus a is equal to 0 and of course a go to zero right that should be pretty clear likewise this right here tells you C is equal to zero man it's weird now I have no cure this right here tells you T is equal to negative B and this right here OTR you for P and then P is negative P so minus P right here this is this part minus B has to finally equal to a number so you see here we have three P is equal to one Oh shadow to simply one prong look at that and then this is equal to P is equal to one-third P is equal to 1 so that means D is equal to negative 1/3 not so bad okay now so bye guys alright so a is zero and this is zero so this is out and all that stuff so let me just write it down this is 0 B here's 1/3 C zero and this is positive 1/3 right so some pretty low partial fractions now come back here and now let's go ahead and do the inverse applause well let's do the first one which is just 1/3 over as to the second power mine plus 1 once they're on the top s square plus 1 right and if you were like um close that and then the second fraction is this which we just add in first now plus that's a whole one that's the whole boil so plus the inverse Laplace we have a 1 sir yeah so we can actually put that in the front right here and wait once negative this register negative 1 P is negative 1 yeah negative answer so this is technically a minus 1 sir yeah maybe to be a negative answer and then we have that over the s square plus 4 so radius s square plus 2 squared like that Allen times 1 all right as usual perhaps I'll show you guys my secret weapon yes I found my green pen this right here it's pretty nice to do this right here is not on the top I need to have a - because this is the - so Adam 2 multiplied by 2 and then divide it by 2 like that yep alright so if you would like you can put a 1 so in the front or the first one so that's ready done like this one and then the ones they're all the Winnifred if that kind of bothers you all right now here we go finally here's the Enzo alright here is the answer 1 start this right here is the sign right so it's the sign and then people use one so just sign T next this end is minus 1 over 6 and then we have sine of 2t just like that done ladies and gentlemen some partial fractions business it's not so bad if you choose kind of too quickly all that stuff in that yeah if you would like you can also do the convolution theorem that's it anyway that's it so halfway done halfway done all right now number 13 right here in first Laplace of one over s to the fourth power e to the 10 s power now as you can see we have two e to the e some number s this is what we have to do first let me just rewrite this for you guys this right here is the inverse Laplace and then this right here is e to the negative 10 s over s to the fourth power like this right and now this is what you can do if you don't want to just a great on the formula every single time here is how you can do it in your head right whenever you see e to the negative 10 as a number s you have to get the unit step function ready when you go back to the T world so to do that let's just go ahead and put that down right here all the way at the end when you have e to the negative 10 s you actually have to have you are t minus this right here's day right it's negative is nice for that so t minus 10 it's pretty much wherever you see this wherever you get right here yeah like this yeah you always have to have that and then what you need to do is you have to figure out the f of T or F of T so T well let me just read it down right here the interest applause if we didn't have we if we didn't have this it's just 1 over yes to the first power all right one right here is three so I need to have a stupid Tory on the top so let's go ahead multiply the top by 3 factorial and divided the bottom by 3 factorial times buttons do factorial right so we end up with 1 over 6 T to the third power so that's how you'll get originally this right here is our F of T that when we go back we have to have F of t minus 10 they improve have to be the same so this right here has to be F of t minus 10 and all you have to do is in started heat you'd write es t minus 10 so finally 1 over 6 t minus 10 and then to the third power just like that yeah so that's it so I think this is the easiest way to do it just how to remember the formula that we're using in this case is when you have the Laplace transform of f of t minus P unit step unit step function unit step function of the same input t minus a naught minus P you might say same thing doesn't really matter just some variables this right here gives you Y to the negative a s and then you have the Laplace transform of the book so you just have to multiply by this guy so that's the deal right there right what's I'm not use green not anything against Greinke all right so that's pretty much it now next number 14 this one we have in first the path of interest and yet it was the first intention of one over s weird this is not fun table right away huh well actually there is the entry that gives you this but I will show you guys how they actually work elf on scratch right so suppose you didn't do the one that I demonstrate for equation number something in the previous video so hmm how can we do this though in fact we don't know this too well but we differentiate this is actually not so bad because the derivative of inverse tangent it's actually going to give you a rational expression in the S world for example to work the inverse tangent is 1 over s square plus 1 right and then of course we just have to work our usual to rub the other stuff and that's going to make it work very nicely so you can go ahead and do the derivative but before I do the derivative let me first tell you this expression in first engine of 1 over S this is actually nice T equal to and against you have the inverse tangent by this expression is actually the same as inverse cotangent this right here it's actually the same as inverse cotangent and in my people might be worried about the domain and all that stuff so this right here it's the same as inverse cotangent oh yes yeah okay and again some people might be finding the inverse cotangent to be some other things but just different branch but usually this is okay right because don't want to be too piece of wood state the domain and range of all that stuff all right so you can do it like that and again if you have the interest cotangent sorry sorry done - just cotangent this right here the same as cotangent of s sorry in first tension of one of us this right here it's the same as in first cotangent of s right and again you can just go ahead and approach it this way but you can also - like this this right here it's also the famous PI over 2 this is the cofunction identity spots the interest person of that so this is the famous a power to minus tan inverse tangent of s maybe you guys will be familiar with this dispersion study more than this one right and later on I'll be doing the trick I thought he proves marathon yeah all right anyway whichever you want to do care as know so you know how to delete a robe he you're good to go now here is why I want you guys to remember so this is the one that we have to use right not what's the difference between note and use well same thing I don't know all right here's deal when you have the in first when you have the Laplace transform of a function that they have a key this right here we know is f of s yeah it's f of s no problem that however sometimes if you have this x just t in that case what you do is in the s world you differentiate this with respect to s and then you multiply by negative like that right well well from this right here in fact I will have to tell you this only works precisely if F of s approaches 0 and s goes to infinity so this is just some little note for you guys because in fact mu prime air show me a counter example if I don't have this condition then this is not entirely true right but with this condition this is true and I will have the video in the description for you guys and that's actually how I met mu prime math mu prime F for the first time so this right here I will just show you two the MU prime math video I think will be in the description for your convenience alright now what we'll do is take the interest on both sides and you will see that the left hand side is just T f of T equals this is negative T TS and then we have F of s right and then you see I have I don't have the Laplace anymore because I would say this right here and then we do the inverse right here so that's a formula to get back to the F but I have little T so it's okay to have a look F of T this right here is just D by DT on both side so we have 1 over T and then we have the inverse Laplace and then you multiply by negative here let's put a negative on the outside yeah so dinky on the outside and then just do the derivative in the S world then you have a better experience with inverse Laplace so this is a very nice formula right mainly very nice Plainville oh no this is not the answer we haven't done this yet here we go right so this is how we can do it to get the inverse Laplace of this right as I said earlier this is pressuring the same as that it's the same as blood and then let me use this version so this is the inverse Laplace of PI over two yeah let me just straighten that PI over 2 minus inverse tangent of s alright good now remember this right here would keep you happy this right here will also keep you a puppy so this is the following right this right here what we'll do is multiply by negative 1 over T so that will have the negative 1 over T first right and then we still have the infrastructure now what we'll do is inside here I'm going to differentiate it in the s world and then we have all that inside so PI over 2 minus in first tangent of s like this so again the idea is that if this expression is too bizarre go ahead differentiate it and then multiply by negative 1 over T now is to give you an answer would still give you the answer for it let's see if this is really any goona here in the front is negative 1 over T and then to the infrastop plus inside here well the rupture of this is just 0 the rift of this right here it's actually negative right this is negative and then we have 1 over this is s square plus 1 s square plus 1 like that and again the truth is if you differentiate this right here differentiate down right here and appreciate his right here they are all the same derivative and again just I'll show you guys the proofs later on all right now we can put an X here on our side so you actually get past the 1 over T now and then the inverse of this right here is just quite sine T isn't it okay negative negative becomes pasty already so this right here finally just give you let's write it down in the classic way signed he over T just like that very very very cool right have a look have a look have a look I have another example for you guys because I know this right here might be super bizarre sometimes so here is another one number 20 no I thought that was number 19 oh wow I'm almost done no number 15 so in first Laplace here we have how on front of these oops no in parentheses and then we have the s number 15 in first applause of Ln s square plus 9 over s square plus 1 ok here's the deal if we differentiate Ln again we'll end up with an AK we will end up with a rational expression so that's good we prefer that if you notice all the Laplace pretty much just in terms of s right in terms of in terms of s in the s world as a Russian ice brushing Network there we go so differentiation first it's a quiet year so I'm going to differentiate the inside first and then just make sure on outside we multiply by negative 1 over T so here is how we are going to do it multiplied by negative 1 of the T and then here we have the impressed dollars and then the inside welcome to differentiate this with respect to s then this is pressure McGurk now we're about to take the derivative my suggestion is use natural log property so I will write this down as Ln of s square plus 9 and then minus Ln of s square plus 1 so that's what we have and then I will close that and then I will close this so that's pretty much the work that we have to do so again where you see this property right here and the real quick check is that I forgot to mention which always check this F of s is approaching zero is it well if you take s to be infinity the inside will be approaching 1 yeah good now and 1 yes zero so this works earlier if you put s to be infinity inverse tangent of 1 over infinity is inverse tangent 1 over C 1 of inverse tangent of 0 which is 0 so again this works yeah so just check the diamond all that stuff anyway though let's go ahead to the derivative here is negative 1 over T and then was to have the infrastructures inside out so differentiate this finally we can use the chandu yeah the revenue of this is 1 over s squared plus 9 but the chain rule says we have to multiply by the derivative on the top so we get 2's yeah okay twist like that the review of the top minus the derivative this is a square plus 1 on the bottom and again we have to use the chandu so these derivative of the inside is also twist so let me just put a 2 in blue why I don't know yeah the chain do you actually okay I think ohm is our can read so far so that's good and then this right here seriously otherwise that would totally look like a 25 that's not good so that's what we have okay now we just have to do the la infrastop class of this anything personal plus of that and that's it notice this is going to be the cosine cosine version isn't it very nice for the nine of course you look at this as three squared on the top is s and this is a square so that match already we don't need it as proper right we're going to do anything and of course this is two and all that stuff so here we go we have the negative one over T or the window front put on parenthesis for the result of the infrastop plus 2 is 2 so perhaps already done why not and then s over s squared plus three squared that's going to be cosine of three T like that very good - two years and hotter - right here this is just cosine T done yeah and then distribute if you would like and yeah so you get negative two cosine 3t plus two cosine T or over T like that and we are done like this all right so that's it so sometimes if not plus transform give you some trouble in post-op last transplants give you trouble differentiating that first and then just multiply by negative one about heat like that all right so that's that you and you might be thinking can we integrating that first yes and an example it's coming up pretty soon so they make this your first oh how many because are taking differential equation this semester if so let me know right in fact I am NOT teaching differential equations I'm sorry Brad I'm sorry prior I'm sorry Eddie yeah just were my formal call to students and hopefully everybody is doing we're including you guys who are watching the videos right now alright number 16 number 16 in first surplus of 1 over s square minus 16 like that oh it's just a coin so that 15 16 I didn't prepare that actually it's not just make a heart of one of course as to the fourth power otherwise the partial fraction would be too easy to do right now factor things out can we do this in your head let's just go ahead and do this in our head actually no I'm not that's doing the head we first get s square minus 4 and then s square plus 4 and for the s square minus 4 we can do this as minus 2 s plus 2 and then s square plus 4 unfortunately cannot be factored this so that's what we have for this one I will show you guys partial fraction for it so here we go we have 1 over s minus 2 s minus X plus 2 and then s square plus 4 now this is not so bad now check this out first we have to have a number of as minus 2 and add another number over s plus 2 linear so on top just numbers lastly we must have a quadratic quadratic over a square passport like this so I will just call this to be a be linear on the top so I see S Plus D all right now that's to the cover-up method and that's to the cover-up in our head because top rap is so how to differential equation is the copper up into a head so have a look here too eight years go back to your original you compound the same denominator and how are you going to make s minus 2 equal to 0 s has to be 2 right to put 2 here and put 2 here 2 plus 2 is 4 this is going to be 4 plus 4 which is I almost want you to say 16 no this is 8 and this is 4 so 8 times 4 is 32 so any is 1 over 32 now to figure out P copy this up and you let s equal to negative 2 negative 2 negative 2 this is negative 4 and that's your 8 so we get negative 1 over 32 right so far so good huh now my recommendation to figure out the CNT is just to the usual way or if you like you can plug in as this equal to 0 or s is equal to 1 any s value that you haven't used it yet that's finally work of its equations but again my experience is just to seduce as let's multiply everybody by the lowest common denominator which is the S minus 2's plus 2 and then s square plus 4 all the stuff and then work out the algebra all right the left-hand side we have 1 and then this right here is 1 over 32 this is going to be study piece art because we have a lot of fractions all right this times that this will be also we have this which is s plus 2 s square plus 4 right next minus 1 over 32 s plus 2 okay so so we have s minus 2 s square plus 4 down Wanda so we have plus C S Plus T and just this and that so is I mean just this which is s minus 2 plus 2i oh this just worked out real quick so here is 1 over 32 this end is s to the 3rd power plus 4 s and then plus 2 s squared and then plus 8 am i right yes then the next part is minus 1 over 32 time is pretty much the same thing so we care s third power plus 4 s minus 2 s square minus H yeah and nasty I will actually do have space like this and then lastly I'll just put on plus this is I will multiply this out yeah so we get C s plus T times s square minus 4 can we see yes ok you guys can see this yeah all right now the piece of pie I just I can just multiply this out so we get nothing can be reduced this right here 1 over 32 s sir power actually I don't need to have too many shadow one of us re - why is this so hard I don't know anyway 1 over 3 to s to the third power and then this times I plus 1 over 8 s plus 1 over 16 s square plus 1 over 4 good now I have some - going on so it's - and this and that is 1 over 32 a third power minus 1 over H s and then minus 1 over 16 s Square and then minus 1 over 4 sorry minus minus of P R plus minus minus Rho P R plus and that's lastly though I have to multiply this out yeah so this times this I just pretty much go ahead and do that first yes so we still have the plus C s third power plus C s times negative four negative four CS yes I then plus D s square and then lastly we have the minus 40 like oh this oh my god all right I just need to figure what's the n these are yeah so let's see do we have the constant terms we do here is the constant term and here's a constant term and here is the constant term right yes so based on this we can conclude that the constant term has to be equal to 1 therefore we know 1 over 4 plus 1 over 4 plus negative 4t so it's just minus 4t this right here it should be equal to 1 rub that out this is 1/2 put it to the other side negative 4 T is equal to negative 1/2 D is equal to positive 1 over 8 so hopefully this is right I don't know because that you didn't write that down I only have the final answer so I hope this is right D is equal to 1 over 8 now how about C where C okay okay what's the s yeah so let's see here is the one with s here is the one with s and here's the one with s so this implies the coefficient we have 1 over 8 minus 1 over 8 and then minus war see this right here there's no s on the left-hand side so it has to be equal to zero this is zero so negative 4c equals zero C has to be equal to zero okay done see you thorough P is 1 over 8 all right now come back here and then do the infrastop pass in first lap class first fraction we have the 1 over 32 let me just read it down like I don't even know space for that mmm one of us 3 to them in just put on your friend I guess so here we go to the interest not part of this which is 1 over 32 all the way in the front first and then in first laplace Oh 1 over s minus 2 so that's good and then just continue the next part is minus 1 over 32 so we have the minus 1 over 32 and then the inverse Laplace Oh 1 over s plus 2 and lastly this is out so we have 1 over 8 and all that so we have to add 1 of 8 and we add one of 8 infrastop class of a member which is just 1 because I put a 1 over in front already and that's the s square plus 2 square like so right now finally we'll see this right here is just 1 over 32 good this right here is e to the 2t good e to the 2t remember s minus a and then e to the HT eat then minus 1 over 32 and then this is e to the negative 2t for the second one last IDO well this is the same version and as you can see this right here we have a 2 so we need to have a 2 right here and that means what / - yeah so it's just like that therefore we actually get 1 over 16 on the outside and then we have the sine of 2t who took awhile and oh I have the minus plus okay yes d- o positive oh I'm sorry T is negative have a look because this is 1/2 so this one here's one and this rupiah 1/2 yeah because 1/2 minus o should be 1/2 and then device or tea shop is negative 1 over 8 so T right here should be negative 1 of 8 and see does it matter is 0 anyway so this right here should have been a negative 1 over 8 and then this right here should have been minus 1 over 16 in ANOVA so oh yeah just like this done so have a look have a look have a look all right so I have to erase all that yeah and now remember this question here this is the inverse Laplace of 1 over X to the first power minus 16 huh right that was what we just did now number 17 I will put it down here and we have this right here which is the inverse Laplace and we have s to the third power looks as to the third power over as to the fourth power minus 16 again okay and this is square [Music] well do I want to do a partial fraction again I don't think so but if you look at this and look at that what's the connection depends how you look at it yeah if you differentiate this the function part will give you that we can also look at this as if you integrate this the function part will be something like this right so depends on you want to look at it so have a look of the following right so I'll tell you guys the answer it's actually very very simple bubbles explain that to you guys in fact can make good use of this for this question that's why I have erased that down yet alright so Rico this is the one now we are actually going to use when we have the Laplace and this is the original Laplace of F of T this right here is just okay look at what the f of S is yeah just like that and if you multiply this by T all you have to do is differentiate this with respect to s and then you multiply by a negative so that's pretty much it now of course again this is only true if f of s goes to actually this is true but the next one is not one hour after this is the 100 oh I can take the inverse on both sides and that means I can put this to the other side I will tell you let's do it like this if you have T f of T this right here I can say it's the same as a to the negative two here and then I'll do the inverse Laplace of this expression which is TDS of F of s ah this is true so that's exactly what we have to do and again this is true if F of s is approaching zero when s is going to infinity right now how we're going to make use of that oh just look at this so from this we know that this is what we know already it's already done one more time for you guys we know that the inverse Laplace of 1 over X to the first power minus 16 this right here it's nice to equal to all that which is 1 over 32 e to the 2t minus 1 over 32 e to the 2t negative 2t and then minus 1 over 16 sine of 2t right ok better better maybe ok all right now what I'm gonna do this right here for the inside I'm just going to differentiate this OOP snacks SSS I'm going to Frenchy this with respect to s now and if I differentiate this with respect to s oh I had to do this on the right hand side just go ahead and multiply by negative T right just go ahead and multiply by negative T with a whole expression like that and that's pretty much it so have a look have a look have a look the let's see just write this down oops SSS D D s right look at this as s to the fourth power minus 16 to the negative 1 power yeah so have a look at the left hand side the following if it will impress that plus 2 litre property first you bring the power to the front which is negative and then minus 1 so you have s to the fourth minus 16 to the negative 2 but don't forget the chandu so multiply by the derivative inside which is 4 s to the third power 1/2 PI / 4 s to the third power and that's the chandu yeah so you can differentiate that and now on the right hand side I'm not going to multiply anything out I'll just keep it this is negative G of all this expression right then as you can see this is almost the same as that example we have the negative 1 over we have a negative 4 so I have to do this just divide both sides by negative 1 over 4 let's divide this by negative just divide it by negative 4 divided by negative 4 so this and now I can so finally I can show you the inverse of s to the third power over s to the fourth power minus 16 to the second power this is nicely equal to this this and that well they reduce to past Deepti over 4 and then all this just cut to write it down again so just copy that down 1 over 32 e - t minus 1 over 32 e negative 2t add a minus 1 over 16 sine 2t like that and this right here is the answer right this right here is the answer for that so pretty cool stuff all you have to do is just recognize the connection between the one that you're doing with the one that you have done already and usually you'll find of good surprise very good whoo all right so I'm gonna have a look have a look all right whoo good a pitch is done and then just one more page check this out I have differentiated questions for you guys that's gonna take a while hopefully not so bad all right so this is the one promise which you guys with the convolution so here we go number 18 so here we have none none under green mallet here we have the inverse Laplace of 1 over s to the fourth + 4 s square plus 4 like this you can factor the bottom and I'll show you this right here it's in fact the same as the interest applause of 1 over s square plus 2 and then square like this now here is the deal I can look at this s 1 over s square plus 2 times 1 over X plus 2 no breaking apart with convolution double regular multiplication so here this is going to be the inverse Laplace of 1 over s square plus 2 convoluted with the inverse da plus 1 over s square plus 2 like you see in a gap and actually let me just rewrite here so I have more space for the other stuff so sorry 1 over s square plus 2 composition in the first class of 1 over s square plus 2 right what's this sign of something yeah I didn't we don't just want just to give you guys some more flavor this is the same as square root of 2 square so on the top I need to have the square root of 2 likewise depay depay that same thing square root of 2 and then divide it by that so for the first one we actually end up with 1 over square root of 2 sine of square root of 2 and you have to do the convolution of 1 over square root of 2 sang of square root of 2 T like this now here we go this is the convolution formula first constant chance watch that indentures would say this right here is my F of T and this is just F of T and I'll say this right here is the G of T right so the convolution former says you actually end up with integral going from 0 to T and you write down F off right so that's right on the F first which is 1 over 1 over square root of 2 sang of square root of 2 you'd write down F of t minus V so this T right here actually becomes t minus V like that so this is actually T F of t minus V first part and then for the second part you actually happen nice one again this is our G of T when you put inside here you just write on govt so just have to change that heat of V so you actually end up 1 over square root of 2 sine of square root of 2 be like this one thing here is that inside it's just a regular multiplication and we are in the v-world so be really careful with that now here is the deal green things are before we do the work here 1 over square root 2 1 over square 2 multiply to the outside so we actually get 1 over 2 and just to show you guys purple this is the purple 1 over square root 1 over 2 anyway here we have integral from 0 to T and this is sign of distributed square root of 2t minus square root of 2 and then times sine of square root of 2t like so yeah now you might be wondering how in the world can we integrate that this is the square root of 2 can you guys see I just realized there's a clear maybe how can I plot that like this a little bit um better a little bit right I'm I am sorry for the lighting anyway we have to integrate this guy right here in the free world but the thing is that you can use integration by parts one what else come two identities that's the one we're going to do so here is the deal already down here for you guys so let's code this to P alpha well just capitulate because why not and this is B sine a times sine B is equal to the following I'll just write this down again for you guys this right here is equal to 1/2 cosine of a minus B and then you - right - you have cosine of a plus B like this right so sine a times sine P is equal to cosine cosine and then this right here is minus and all that stuff right now just worked out the integral I hope you get so we have the 1/2 multiplied with that so we'll end up with 1 over 4 all the way in the front integral from 0 to p 0 to t I'm sorry and then the first one we will get cosine off we have to do a minus B so this going to be square root of 2t minus square root of 2 P minus P which is that square root of 2 P right so that's the first part and then we have to - and again i factor out the 2 already 1/2 already and the next part is cosine of and just add cosine of a plus B which is square root of 2t minus square root of 2 B plus square root of 2 B like this all right TV now all the way in the front we have one over four so this is integral from zero to T and this is cosine this and that can be combined so we have square root of 2t minus two square root of two fee and this and that can be cancelled yet so we get minus cosine square root of 2t DT who we are ready to integrate so here we go we have the one over four in the front put down the parenthesis for the result integration integral of cosine is past this side so that's good so we have to pass this sign right here and the input is the same so let me just stay square root of 2t minus 2 square root of 2t and the reason we can do this it's because inside linear now when you do this don't forget to do the dual chain then get in reverse chain rule we are in the v-world so the interrupt here of this part is this so divided by that so I will put down 1 over negative 2 square root of 2 yeah right so that's the antiderivative of that for the second one now it's trickier because you have T right here however we are in the v-world so all you have to do for this is just write down minus cosine of square root of T school of 2t and then you just multiply this by V because again we are in the v-world so that's pretty much it now we just have to plug in plug in so let's see we have 1 over 4 all the way in the front man let me see how I we have the 1 over 4 in the front okay clogging T into V right plugging T into feet plugging T into V so first part we get up for this time to do house a yeah mm I'll see what he puts on to blue so this part is let's just do that sorry anyway go this is negative this is positive 1 over negative 2 squared we up to and then plugging T into V so we have sine of square root of 2t minus 2 square root 2 this becomes T now so that's good and then we have minus cosine square root of 2t and again this is a T now right so that's the first part and then we're plugging 0 so at a - plug in 0 so we get 1 over negative 2 square root of 2 sine square root of 2t minus 2 square root of 2 times 0 and then minus cosine square root of 2t times 0 right so it has twists persons inside now finally we have the 1 over 4 and that's good a blue pen to simplify we will see this and that is equal to negative write this and that can become pious negative square root of 2t so that's good this is just that doesn't really do much this is out this is out right now here's the deal here's negative instead of the sign but signs an odd function so you can put it on the outside so the first part is going to be positive so we get positive 1 over 2 square root of 2 and then we have sine of square root of 2t yes and then this is minus cosine square root of 2t and he and usually we could put on what he first so let me put on this G first make yourself more clear and then for this part is just minus minus is plus so we just add right again just add 1 over 2 square root of 2 sine of square root of 2t like oh this yes we can see good now let's see another thing we can do is this and that can be combined yet and when you combine this and that is just this and that is just that too old for that so the torque become all right and then that's pretty much it so I will just write this down this is 1 over 4 I would multiply this in square root 2 right again this times this plus da is 2 over 2 square root 2/2 and to Kenzo so rad the square root 2 that on the bottom and then 1 over 4 in front like this and then this power which is sine of 2 sine of square root of 2t man who wants to do the convolution man I don't know anyway and then do this is minus 1 over 4 T cosine square block 2 T like this and then we are done yeah and secured rationalize the denominator power I'm not gonna do it right here because I deal a lot already oh you guys so have a look out for the public so just like that all right next question number 19 right here originally I'm debating to see which one to show you this one just you factored and into partial fraction by thin study we tell you a lot of partial fractions already so not so much so actually I'm just getting to how to stop differential equation which start first transforms yeah Zoe let's go ahead do that and then we'll rearrange the queer I'll just yeah I was just doing this water then so question number 19 so it's this one I have Y Prime and then - no sorry plus 2y equals sine of 3t and then we also have to have the initial condition Y of 0 is equal to 0 and if you wonder why if we don't have an initial condition well just go back to the presence and hey where's my initial condition man anyway this is how we're going to do it first you apply the Laplace transport to the equation so go ahead you can just do it like this and this is the original what you do is this Laplace transform you will take this equation in the T world to the S world and then just mess around over there I just do the algebra over there and once you get it done take the inverse Laplace and we'll get back to the T world that's it let me show you first thing first the Laplace transform of Y prime is the following uks times and in this case let me just make a note right here if we take the Laplace transform Y of T which is just the Y function in the few world once I'm in the 8th world already as Y of s well word US Capital y of s all right here we go so is little s times Capital y of s right and then you - the initial condition which is why op 0 and this case is here also I'll put on see right here and again this right here is just Y of 0 yes as Y of s - well she rope so this is what we have for the first one next take the Laplace up to Y we get plus 2 Capital y of s like that next we have this you get this right here it's the p value which is 3 oops you shouldn't be great this is the 3 so I have the 3 right here over s square plus 3 square just not like that alright so that's pretty much the first step now what you want to do is you want to stop work well wire pass is that's why I said you have to do the algebra work in the s world and then once you are done take any person can come back to the t world all right so this is not so bad because you see only these two terms yeah so we have s plus 2 times y of s which is equal to that namely is 3 over s square plus 9 like this and of course we can just divide on both sides so we are looking at Y of s it's equal to 3 over s plus 2 and then s squared plus 9 like this now we will take the inverse Laplace both sides and then we just write it like this first up last down everything so I'm sorry to practice not a nicest thing I cannot draw the brackets the brace is actually all right the left-hand side will keep us the little f of T so Y up here I'm saying why is it lettuce is going to give us the little wild and now we just had to work that out and you'll see this is why I didn't want to do the partial fractions because we start to do a lot right here so partial fraction thing actually - right here so we have 3 over s plus 2 and then s square plus 9 so - one more for you guys yeah I see first one is s plus 2 some I'm brown on top and then next one s square plus 9 is B s C now to figure out a go back to the original copy this app and you ask yourself how can you make s plus 2 equal to 0 s has to be negative to put here negative u square is 4 plus 9 13 so aes3 over 13 now for the B and the C to the traditional way multiply everybody by the denominator so I will end up with 3 by itself to this cube chocolate so you have 3 over 13 you would have to multiply by this s square plus 9 and then plus B s + c x as pursuit yeah and then just work this out this is 3 over 13 Square and that's + 27 over 30 yes Plus this and that is be s square plus 2 PS plus CS plus 2 C good now check this out here we have let's see here we have oh man here we have this yeah 13 that's just ready done here we have this 1303 over 13 s square and all that so say 3 over 13 plus B and of course this is the square term and then plus this and that is the S term so we have to B plus C that's the S term lastly we are plus 27 over 13 plus 2c like this that's the constant term which one is the easiest to do well pretty easy both this right here has to be hot 0 because we do not have the s squared on the left hand side so that means B is equal to negative 3 over 13 good yes and then in that case what's sido well I will hum no I'll do here because I have the confusion I don't like that so here to be which is that negative 3 over 13 plus C has to be consumer as well so this is going to be passed these six on the dollar hands I see you go to pass 6 over 13 like that so that's not bad in my opinion yeah not so bad in my opinion yeah okay all right so let me just push it down together first one here we have to do the Laplace infrastop ask is 3 over 13 and then we have to do the interest applause of 1 over s plus 2 yeah let's just look bad now P is negative 3 over 13 so we have 2 minus 3 over 13 and then inverse Laplace and that's with s so we have the s over s plus s square plus 9 which is the same as 3 squared right and then of course make this longer C is 6 over 13 so I will just put down plus 6 over 13 and then I will just erase all this cuz we have all that already yeah 6 over 13 and then we have the inverse now class of these six over 13 yeah well lucky I put that down already before you race anyway 1 over s square plus 3 square nah all right now ladies and gentlemen first equation by first answer right the first differential equation with stop plus is equal to Q over 13 this is e to the negative 2t and then this is good it's the cosine so it will have 2 t5 whatsoever minus 3 over 13 cosine of 3t this right here you do have to do something here is 3 so Cohan multiply by 3 and divide it by 3 and you can reduce of course so just reduce so just a 2 so 2 of the 13 sine of 3t and doing to put on plus C no we don't the beauty of doing this is that the constants were soft throughout the procedure already so it's not so bad right so very good stuff right the last question number 19 alright question number 20 next we have the second eruptive 1 so Y double prime minus no why and that's equal to 3 T Square and we have Y of 0 is equal to 2 and y prime of 0 is equal to 1 all right here we go go ahead into the lab class this is just the original version now first this right here we begin with s square so it's crazy already ok s square and then we get Y of s right a plus of the second term of T is y square it's s square times y of s and then minus s of s times Y of 0 so as x - so this right here is y of 0 limiting its a y of 0 or y prime 0 first man it was y all right to show you how we did that so the plastron small F prime this right here is equal to s now class transform of F and then minus F of 0 yeah just just like yeah like that and then now I want to have the second derivative so I will just put F prime here then s and this is f prime minus F prime almost right yeah because this right here is just what we have here so just put this down so as s double prime yes I'm right here yeah and the minus F prime of s my minus minus this which is 1 this is right so that's the second derivative that's the first part then you continue - for capital y of s and that's uncool to do the Laplace of here man you have the three times and these two so you do two factorial over s 2 plus one why don't you know anyway let's go ahead and do this stop complaining all right this and that has to be go to the other side and this and that or just factory so we have s square minus 4 and then capital y of s ok and then this and now go to the other side so we have 1 plus 2 s and this and that three times that is 6 over s to the third power like this right and then we have to define everybody by the s square plus s square minus 4 so Y of s is equal to 1 over s square minus 4 serious due to them any partial fractions I'll do partial fractions on one of them so this 1 1 minus 2 s 1 plus 2 s over s square minus 4 minus 4 and then plus 6 over as their power s square minus 4 like this why why do I do this to myself all right that's see this partial fraction was gonna do in your head y up s is equal to the following this right here I need to have s minus 2 and plus s plus 2 now check this out to make them straight down as minus 2's plus 2 to get this number you have to cover up and let s equal to ya put it right here so you have 4 plus 1 is 5 over 4 so it's 5 over 4 so let me just put this down right here PI over 4 and then here is s minus s plus 2 so you cover this up you there as equal to negative 2 so you can negative 4 about it then you put negative 4 so you get so you put negative 2 so again negative 4 and then 1 plus so it's negative 3 yeah so it's like that oh man seriously and now to to a lot of partial fractions for this man y-yeah I do that myself keep this one I don't know guess my lot Oh should I not do it all right ok ok I'll do it alright so for this right here let's see so then this is going to be this six over s-sir power as - so six over s third power s - 2 s plus 2 last year and way that we can but - it's gonna be basalt regardless another way we can do is just combine it as a big fraction person it's just okay how do that I was seriously - that man all right I have to really make sure I can finish this before they write anyway so this right here is divided by Thalia and you have the that so I watched how to get the S to a third power right here so the mean just write this down defy this bias twister power multiply that alright so again you will get the following and now if I everybody here so what you will carry is this and I which is actually this end is to as to the fourth power and then plus s to the third power and then plastics right so just get the commenting on all the stuff and then I will just divide that right here as well so we have to ask to a circle here and then divided by our choice s minus 2 and then s plus 2 okay okay it's a horrible partial fraction so 2 s 4 plus s circle right this is Esther power that's s to a 3rd power plus 6 all right and then this is a third power as minus 2 s plus 2 now the horrible thing in which this is that you got to build other powers so first you actually have to somehow know something over s plus next one something else over s square and then plus next masti over as to the third power and then you continue by doing this plus T over s minus 2 plus Yi over s plus 2 all right good news drew up them we can use copper up so let's do C first curse foresee all we had to do this go back to the original and then cover this up so already stop see it's equal to when as this equal to 0 so you get on the table it's actually six over this is up right so you have negative two times two which is going to be this is negative three over two good now when you have T that's when s is equal to two so put a 2 into all that man put a 2 right here is don't put the Torah here this is 8 times 4 and then put a tool right here this is 16 that's 32 plus 8 and then that's 46 so that's 32 46 over 32 + / 2 23 over 16 23 over 16 does not look right it should be stood over 16 what the heck this is just not for you oh my goodness you know what I wrote down the wrong answer I wrote on the wrong question so whilst you wrote oh yeah no the white primer should be negative for this right here should have been negative oh I'm so sorry this right here show up in negative 4 so it matched with my answer key so this right here show up in negative 4 so it becomes plus 4 so if you put a full ride to the other side this will happen at negative 4 and this one being a minus 4s I'm so sorry and then this right here should be a minus 4 yes all right let's do again all right see it's equal to when s is equal to 0 I'm not that was all wrong it was not 16 was this 6 anyway what you see anyway see is when s is 0 so you have 6 over negative 2 times 2 and that's negative 3 over 2 so that's good yes that'll work all right so that works I hope yes now P is when s is equal to 2 you put a 2 into here into here in here Stassi - in here this is a 16 that's 32 yeah and then 2 here's eight that's 32 so it's zero so you have six on top over you put two here's eight you put two in here it's going to be four so you divide by 2 so 3 over 2 16 good all right lastly Yi is equal to when s is equal to negative 2 what is equals negative two here and you put it on the top this is 16 times 2 is 32 and then negative 8 32 32 so it's a 64 and then plus that is 70 over you put a negative 2 in here is negative 8 times you put it eyes this negative 4 all right yeah and then divided by 2 man is 32 over not 35 over to 1635 over 16 okay so far so good at matching my answer key now so so far so good hahaha let's see what a and B then God I really don't want alright let's just do it hey you can do it like this okay you see how we have just s all these numbers yeah this this numbers are the s numbers are used so what you can do is you can just that as equal to PI over L that you want of this s is equal to one and not as twisting ahead for one here this is 2 and that's minus 2 and then we have that is for yes good one negative one and three so just like that so that's one s so a is I don't know actually a all for that plus P over 1 square just a so doesn't matter and then C is that so it becomes minus 3 over 2 and then P becomes d is this yeah well as this one so it becomes minus so remember 1 minus 2 is negative 1 and D is there so it's minus 3 over 16 one in here is 3 3 put it here so it becomes plus equal to start 35 over 16 but you have a street like this man I'm not making this is just not fun to multiply by heat negative 24 liquidy 24 so negative 27 over sixteen perhaps nakata pi/3 so it's negative 81 over that so negative 81 plus 35 negative 81 plus 35 man 46 now yes negative 46 46 so it's negative 46 over negative 46 over 3 times 16 god this is just nothing to do okay I will let you guess we call the rest okay I think according to the answer key that half that right here I need to save some time because I need to finish the other questions so let's see that see that see this is done that's done and this is also done and now they are read for sure so this right here is keeping that constant so this a show have been negative is to operate it should happen it should it should be negative to operate and then the B you should be 0 because there's only four terms on the answer so it should be zero so if you check you'll see that right oh it is B 0 let's see negative I don't think so something's wrong I don't know man but just go ahead and do it on your own all right all right so the struggle is super real anyway a is negative 2 over 8 s so negative 3 over 8 and then you have to ask right here so first and then p0c use that so C is this so it's minus 3 over 2 and then you have the 1 over as twister power 1 over s to the third power and then D is 3 over 16 so you just put us through over 16 3 over 16 and then you have the 1 over s minus 2 men a lot of work as - - and last year you have the equal use 35 over 16 plus 35 over 16 and you have the 1 over s plus 2 all right ladies and gentleman I'm so sorry about this but here we go take the inverse Laplace and everybody now right so just calm down a little bit I know I'm the only one who second nervous so first one you get is negative 3 over 8 next one notice that the s the n is equal to 2 so you actually get 2 you have to multiply 2 factorial and divided by 2 factorial so you actually get negative 3 over 4 and then T Square and then plus 3 over 16 E is 2 so it's 2 T and then plus 35 over 16 eat with it negative 2t like this all right so just like that yeah hopefully your teacher that you use another method to solve this because this is actually not the best method to solve this particular differential equation you might be easier if you use the empty terminal coefficient I think I'll do that for the next differential equation mirrors also the higher water differential equations all right all right so 21 y double prime minus 2 y prime minus 15y equals 0 Y of 0 equals negative 2 y prime of 0 equals 1 so these are what we have now go ahead and get work la plus in action now this right here is s Y of s minus we have no sorry s Square and that and these s times y of 0 first future is negative 2 so don't make it too again this is y of 0 and again s you're being red and then minus this which is 1 so this is y prime o 0 so that's the first part next we have the minus 2 so we have to do - to Laplace transform Y prime is s and then Y of s and there - right but it was a minus 2 so I have to do - - and then the - and then we had to multiply by what x y of 0 which is negative 2 so the can you see is the y of 0 right and then minus 15y of s like so equal to 0 all right not so bad hopefully not all right we have this this and that the other ones have the Y of s so we can factor it out we have s square minus 2 s and then minus 15 and if you notice this right here it's actually from the original very cool and then you had a y of s and I'll put the other things to the other side so this is negative 1 this is minus 3 so which is minus 5 so the minus 1 minus 4 which is minus 5 put it to the other side five and so we have negative 2 s plus 5 now see this is why I had to write yes all right so this is y of s equals negative 2 s plus 5 over that factored out already which is s minus 5 s plus 3 like this all right so of course I can just do the partial fraction right here for you guys this is nice and easy s minus 5 plus sign of the number and this is s plus 3 to figure this number go back to this cover this up s has to be 5 put it here negative 10 plus 5 is negative 5 I promise I didn't do this on purpose that you know there were so many fights all right I'm sorry so if negative 5 negative 5 and then put a negative put a 5 here so it's negative 5 operate now for this number Cobra here s has to be negative 3 put it here is negative 6 so it's positive 6 times past is 11 over negative 3 right over there is negative 8 like this yeah so that's pretty much it now I will tell you let's go ahead and do the inverse Laplace so I work here Y of T is equal to here we have negative file pay that is great negative 5/8 and this is e to the 5 key and a minus 11 over 8 e to the negative 3 T much better huh yeah much better than your earlier one so that's it that's it that's it and this is also a way to show why when you have this kind of since you and out east e to the Y over but yeah ok number what that was 21 nice 22 is number 22 we have y double prime plus 6y now 16 y 16 y and that's equal to cosine of T and we have Y of 0 equal to 0 and we have Y prime of 0 is equal 1 all right so now go ahead and do the Laplace here we go for the first one we have s Y so esta s squared for the first one we have s square Y of s minus s times this which is going to be 0 that's nice - da which is minus 1 great that's the second degree of you right to the positive second derivative and minus 16 plus 16 and then we have died is just Y of s and that's going to be we have s on top and then this is a square plus 4 squared so that's pretty much it it's not these are plants that's bothering you it's the partial fraction that's going party UI anyway that's out this is one will be on to the other side so you know factored out s square plus 16 and then we have y of s so that's good one bring that to the other side so we have one plus s over s square plus sixteen like this okay defy everybody by that so we get y of s equals one over s square plus 16 plus s over s square plus sixteen square like this this right here is not so bad so I will just go ahead and write this down for you guys already perhaps our cubicles on some Christmas color why not if the Laplace is in red then interest our pass should have been green by anyway that is one time Y of T this right here it's going to be well well I need to have a 4 so x 4 divided by the four so because this is 4 squared so we get 1 over 4 sine of 40 very good stuff yeah now but this one though well you have to do the scenes that we did the root key of this will actually keep you guys something that so check this out so if you have the Laplace inverse if you differentiate this with respect to s of that which is s square plus 16 to the negative 1 right well if you would like complain negative right here and in our case you just duty of F of T G of f of T right which is this is supposed to be F of s and then this is supposed to be 2 F of T so T yeah but if you know this which is sign we would gather so I will put this down with yes first one over for saengil court I believe we have the one over four that's correct I hope yes yeah it's correct all right so if you do the derivative this regular when you're on minus one and then to the chin do so you do get the inverse Laplace the revenue of that is going to be s square plus 16 and then you square that and then if you look at the chin do you have that 2's it's equal to this which is T over 4 sine of 40 man doesn't this look bleep really pretty really pretty yes he does divide both sides by 2 so you get this right here this plus oops this right here it is plus and I put down as 1 over 8 divide the 2 right here so we get one of H and T sine of work huh not so bad if you know how to do the infrastop class do you the quick way sometimes yeah just like that right alright alright alright two more to go number 24 now on number 23 sorry cuz I changed the question here we have y double prime plus y equals the unit step function and we have the t minus 3 and we have Y of 0 is equal to 0 and then Y prime of 0 if we quote you one now here is a deal in first I mean original apostle of course now let's go ahead and get work right here we can just nicely write down our thing which is s square Y of s minus s times y of 0 which is 0 and a minus y prime 0 is 1 and then you continue you add y of s because this is just Y so this is well this in yes world and now you have to remember what's the unit step I'm showing the s world well it's going to be something over s and that's e to the negative a yes so it's negative 3 like that right so that's pretty much it not go ahead and just do the work this is out so do you have to want to put it to the other side so we have the s square plus 1 and then we have the y of s that's equal to 1 plus this guy which is e to the negative 3 s over s T by everybody by that guy so we get Y of s equal to 1 over s square plus 1 and we add e to the negative 3 s over s times s square plus 1 like this right alright so here now we can just do the infrastop class oops con call up a fiber here I'll just do this right here first Yop this way I see good watch sign yeah pretty good sign well for this one it's not so easy we have to do the partial fractions again what she had told us so many ways partial fractions man for this one I would take some time to do it what you do is ignore that e to the negative 3 T true yes for now right so just focus on this part so I will just write this down for you guys so note 1 over s let me just write on everything blue and no more I'll still keep it I will still keep my challenge 1 over s times s square plus 1 this is equal to 1 over s plus no no no no it's not one of us it's one of us but they will show you anyway a BS + e to become a you put this to be zero so a is equal to 1 so that's good now let's see if I would get this right you multiply everybody by the denominator so multiply by everybody by s s square plus 1 this multiply we get 1 multiply this is just s square plus 1 multiply which is gear plus distribute the S yeah so we get be s square plus CES just algebra and as we can see this is B plus 1 s square this is plus C s and M plus 1 1 match already so you can see B plus 1 has to be 0 P is equal to negative 1 and C has to be so that's that year now for this one this is how you have to do it for this yeah you actually have to do this first that's doing first a pass of the of the pot without the e to the negative yes first we have one over s and then B is negative one so it becomes minus s over s square plus 1 so this is the part without e to the negative which is Riyaz first and if you do that this right here is just 1 and this right here is just cosine T so this right here is here and our code is to be our f of little T right why because whenever you have e to the negative 3s the S right here you will get the unit step function as well and you will be unit step function of t minus 3 like that so we will have this as the factor well we go back what you have to do is this portion has to be f of t minus 3 so be sure you put the t minus 3 into this T so we get 1 minus cosine t minus 3 like that right so that's the answer it wasn't the worst one the worst one was the one that we struggled it because of the partial fractions but this is right here see this right here is it have a look ok ladies and gentlemen there is the last one okay my hands are always so dirty all the markers men's like not the greatest marker that I'm using okay y double prime minus y prime and then plus 5y this right here at tu 0 and y of 0 equals 1 y prime of 0 it's equal to 2 I really want to do this one because if you do it with the usual way meaning if you that Y as e to the RT and then you get a characteristic equation you get complex root for that with the plus it will explain why you end up with the answer like that so that's do it first I already know the pass Tapatio here we go y double prime we get s square Y of s minus next is s times y of 0 which is 1 and the minus 2 right so this is the second derivative portion now we have a negative 4 so we have negative 4 right here and then for the y prime we get s y of s and then it's going to be a minus but I will have a I will have to distribute the negative 4 so it becomes plus 4 times F of plus times F of a times y of 0 4 times 1 like that lastly 5 and then this is y of s and then the best thing is that the interest lab class of the world allow original path of spiritual 0 so have a look now let's just do this all right so here you see that this part this part and that part they all have the Y of s and the other ones they are just numbers actually which is no there are not just numbers anyway s square minus 4 s plus 5 and then you have the y of s right after we factor all that now this is minus 2 this is 4 so as to bring it to the other side becomes negative 2 negative here it's actually past yes and then - - yeah so - thank you head and finally you can just go ahead and divide that so we have Y of Y of s equals s over s minus 2 over that denominator guess what we have to complete the square so let's do this right now come please queer I'm actually not completed square we get s minus 2 Tarragona s minus 2 square plus 1 this is so good isn't it again complete the square in your head all right huh what's the answer for this one but if you of this already let me just write this down for you guys finally our key because the green for the inverse Laplace this and they are match like very nicely huh so let me just tell you why op t this is the one with s so you get a cosine version so you have the cosine and this is one square so this is just 1 yeah so just cosine and you just had multiplied by e to the a is 2 into the 2 T times cosine T and we are done whoo this is the answer at the end but it all right so Wow all right so the only question that I was not able to finish completely was question number 21 because I study make you study too hard yeah by the way hopefully you guys will find all this to be helpful again 24 inverse Laplace transforms for you guys this is the interest not plus marathon and let me know what you guys wait to see next right and of course sure it gets my medal this is a smaller one but this is the one from 2005 this is actually my first one that I actually trained for it my 20 my 2004 it took me six hours 30 minutes because I didn't really train for it but this one I did and the time was like 6 hours I mean the time for this one was 4 hours and 30 minutes something yeah so oops I don't use my right hand because my mind's so dirty all right it was good so this okay hopefully you guys find all this to be helpful again all the links will be in the description for your convenience if you have any questions just let me know this spent a lot of time to prepare all this and then work that out so if you enjoy my video please give me a like thank you so much and as always that's it bye

Channel: blackpenredpen

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Keywords: Inverse laplace transform, inverse laplace marathon, inverse laplace transformation properties, solve differential equation by laplace transform, inverse laplace transform with completing the square, inverse laplace transform with partial fractions, inverse laplace transform examples, inverse laplace transforms, inverse laplace, inverse laplace transform formula, inverse laplace transform of derivatives, inverse laplace transform table, inverse Laplace transform of arctan(1/s)

Id: Gx6yQcmLuH8

Length: 176min 15sec (10575 seconds)

Published: Sat Mar 28 2020