Differential Equations: Final Exam Review

um you all want to stop with the bernoulli yes do Bernoulli so this is two point five number nine so I wrote it down prepared and prepare two point five number 9 so we have dy/dx so you I haven't done this since you've done it this is gonna be fun I just wrote it down like I haven't done it equals I know something stay here yeah I know yeah if I go that far you can't see it right over there you can right here it's good right but over there for some reason yeah yeah gotcha yeah I'm aware of it boom so this one is Bernoulli and you know what's brinly because anyone know why why is it Y to the five yeah whenever you have Y to a power like this you know it's gonna be a Bernoulli so the first thing you want to do is distribute the Y and keep that Y to a power on this side and get everything over here so just tribute the Y so we have dy DX dy DX equals so we'll distribute the Y so Y times this we get X Y to the six the six yeah well I just woke up still waking up and then Y times negative one is negative Y yeah I think it is I haven't I have the problems written down here but I haven't done them so this will be interesting so we're almost there we need to get this on one side by itself so maybe just add the Y to both sides so plus y plus y on the exam you know look for this you know you should you should have a Bernoulli right so this is going to be dy DX dy DX plus y equals x Y to the sixth so now it's like in the standard form right so we have dy DX plus y equals XY to the 6 it looks kind of funky to me but it looks yeah yeah that's what's on my notes yeah I don't need this anymore so does nothing else notice any questions so far ok so now we have to identify N so obviously what would n be in this case six I'm gonna write it way up here so I'll just do this like two separate stuff n is six and the formula is this it's u equals y to the 1 minus n that's what you want to work out next so identify n and then you workout you okay and you just go it's all it's very procedural like once you know the steps and once you can do one of these on your own if you can redo this when you're ready for the bernoulli right there's only two Bernoulli's in the homework I think right that I assigned is like review questions yeah okay so n is 6 so this is y to the 1 minus 6 so it's y to the negative 5 right 1 minus 6 is negative 5 right well my you can solve for y by doing a bunch of work or you can do it in one move by raising both sides to the negative 1 over 5 good Kenny so you just do this like this very good and then negative 1 over 5 beautiful stuff right so these cancel these cancel so I'll go ahead and put the Y here on the left hand side so y equals and just as a warning if you watch the videos that are in the homework the book does it differently so the videos are probably different as well I have not watched the videos I don't I don't know so we're here so the first thing you do is you isolate you know this piece by itself identifier and workout you and then you solve for 4y you solve for y okay so now we have to find you Y DX right so dy DX dy DX so here's the thing you I'm going to write it again up here U is equal to Y to the negative 5 right y is a function of X therefore U is also a function of X right so when we take this derivative we have to use the chain rule right U is our inside function we bring down that negative 1/5 like this then we get you so negative 1/5 minus 1 is negative 1/5 minus 5/5 negative 6/5 that's a chain rule yes that's exactly what it is so don't forget to do your DX it's the derivative of the inside if you're confused just memorize it it's terrible advice but it'll work you'll get the right answer so because if it's not there you'll get it wrong on the test what evil white there with the d UD accident for the chain can take you and replace it with ax differentiate with X and I you have use a separate function and D here or D X that's the derivative of U with respect to X yeah mm-hmm it's like if you had I'm gonna erase this but I'll write it I'm just gonna erase it say you had x squared plus 5 to the negative 1/5 and you set the derivative you would get this negative 6/5 times the derivative of this if this is U its derivative is 2x that's d u DX yeah totally worth it good yes yes it's worth it most people don't understand they just memorize it which is fine but it's better to understand stuff mm-hmm I like drawing these lines it makes it easier for me to see any questions so far the next step we're just gonna substitute right all right so now we're gonna make a substitution so now we go to this one and we're gonna replace everything so dy/dx is this piece here so this is gonna go here in this piece here okay so it'll be negative one-fifth U to the negative 6/5 D u DX I'm glad I left some room there that's good I didn't do that on purpose but that that worked out Plouffe why yeah you to the negative one-fifth mm-hmm-hmm and then that's equal to x times y to the six so if so it'll be six fifths right because if you to the negative 6/5 because you have y equals U to the negative one-fifth so it'll be U to the negative U to the negative six 6 v 6 v feels like it's easier to see in this room than in the other room right it feels it doesn't does it feel easier I feel like oh you're closer that's what it is because usually like look at the smaller room but it's not what - why yeah that's from this yeah because why is this so when you raised to the sixth power you get ya know it's worth it so you would get that and then 6 times negative one-fifth is negative six days you see it you see it yeah yeah yeah you have to put you have to go back in here and do it let me pause yeah no oh right right that could cause problems yeah go back to this one go back to this one any questions so far it's so far I think you could do it again like maybe in okay alright so now we have to make this a ones you might say all divided you could do that but that's hard so the way I do it is I do this I write down what 1/2 the x to make that a 1 that's just the way I do it I just so multiply and that's how I was doing it very good Kenny right to the negative 5 gets rid of the negative 1/5 and the you to the 6th 5th makes that cancel right because when you add 6 fifths to negative 6/5 you get you to the zero and you to the zero is 1 right so so I usually do this and then I perform the multiplication in our heads so this time is this it cancels so we're left with D u DX it's a d u DX - is it - - 5 u we did it right yes yes if this is not a you game over so let's check that so so 6 fifths plus negative 1/5 is 5/5 right because when you multiply these you add the exponents so 6 fifths plus negative 1/5 is 5/5 that's 1 and this is equal to I forgot what we're doing negative 5x yeah because he's completely canceled right the U to the six fifths and the U to the negative what's beautiful this is kind of fun spin a while since we've done this kind of like this it's kind of exciting step up from series right like even though it's the first the first s is the worst yeah yes so it's based on this very good question so the goal is to make this a 1 so to get rid of the negative one-fifth we're using negative five and then to get rid of this we have to multiply by u to a power so that when we multiply them when we add we get zero so six fifths plus negative six fifths is zero so if it was this let's just say let's pretend let's pretend it was this do u DX then you would do negative 3 u to the 2/3 so you would do that you're adding yes the exponents add so it's 2/3 plus negative 2/3 which is U to the zero now it's worth it it's worth it ask ask ask we're almost there kind of yeah kind of I'm thinking okay I know what to do so now it's linear right now it's linear so now we have to find our integrating factor right and that was eetu the integral of big P and this is always your big P okay so mu of X is equal to e to the integral of negative 5 DX right this is called the integrating factor this is linear you'll also have a linear one on your test so by doing this problem in a sense we're doing two problems in one I almost started with the linear I thought no it's too easy let's start with this one that doesn't mean you shouldn't study the linear ones right make sure you do the review questions like ok huh if you can't you leave you this but still worth doing the review questions trust me like to worth it yeah yeah that's totally worth it so this is gonna be e to the negative 5x so that's our integrating factor and I say it's worth it because you get integrals and stuff like when you're doing the homework you might know the steps but you might get like Levi said some funky integral like and by doing the homework problem you'll be able to do it again on the test so mu of X is equal to this so I'm gonna put this in a box that's not the answer but that's our integrating factor right so now we're going to take this and multiply each of these terms all right multiply the whole equation by e to the negative 5x yeah it's this P it's whatever is here but once you do you know it's always with respect to X in this case because it's DX it's D u DX so if it was like dr d-theta it would be d theta if it was dy DT it would be DT you just take right mm-hmm that's for P of X that variable is X right yeah yeah this variable is X yeah the what's it called the independent one Brian had to think of a fancy name for it okay so we multiply everything by this so this is going to be e to the negative 5x d u DX minus 5 e to the negative 5x u equals oh this problem is super awesome oh yeah tabular on the right yeah you're always gonna end up with something here so this so all we did so far is put this here put this here and then put this here right this piece isaac is revised this this is always magically going to be d DX integrating factor times u so i don't know what when we put first I'll put the integrating factor first so e to the negative 5x times u and that's equal to this we'll check that in a minute so this will always be the case yeah Levi [Music] yeah yeah so we're here now we're gonna check so yeah let's check this let's check this so this is the product rule right so it's the derivative of the first so the derivative of e to the X is e to the X so it'll be e to the negative 5 x times negative 5 that's there times the second plus the first times the derivative of the second let me do it again the derivative of e to the X is e to the X so it's e to the negative 5x times negative 5 boom there it is it's derivative the first times the second plus the first times the derivative of the second alright so it checks you can always check you should always check if you want to you don't have to you can just take a chance and just live dangerously right it's your choice right you can just so now to get rid of the DDX what do we have to do to both sides integrate yea integrate everyone okay with this everyone okay with this piece so it's all you just memorize it it's always mew of x times you because just memorized okay so integrate both sides something's gonna put integrate integrate so when you integrate the left hand side the DDX goes away right so we get e to the negative 5x times u and i'm gonna go ahead and put the integral sign here negative 5x each other that's tabular yeah you can use parts or you can use tabular let's go with tabular just because it's way faster to do tabular it's too early for parts I [Laughter] had breakfast hours ago okay so so let's use tabular so tabular you can use tabular whenever parts works and whenever one of the factors is eventually zero after repeated differentiation so negative 5x is eventually zero if you keep differentiating it so write that piece down I love tabular this is so much right and you just yeah give up no no keep it up again so it's on negative five one more time and you get zero when you integrate this you just keep dividing by negative five yeah you're ready for the test and then get him one fifth even the negative 5x and then one over twenty five remember the thing called partial credit shoes so like you know the further you get on every question but the most important thing is that when you do this question you don't say oh it's a homogenius let me do it that way it's not so that would be bad so take if you take the right approach you get you know a lot of effort points what do you start with plus or minus plus so plus minus plus and then you draw the arrows right you have to become positive I'm always scared to erasing class because then you go home like oh did he erase it like I stab his teacher he would erase with his hands and was really confusing okay so this times this these cancel right you're right it's positive so it's gonna be x times e to the negative five x and then plus again right 1/5 1/5 oh this is so awesome 1/5 even a or 5x now we can add our C so plus our constant C let me pause here it's only been 16 minutes which is okay 17 minutes I don't think we'll be able to do everything I want it to do today cuz no but it's alright sorry I got a list you know let's just yeah yeah I feels right I forgot I have to leave I'm gonna go eat then I have another review after this cup one longer though it's long a review okay so now we're not done right we that replace you with what it is so where is you oh my god it's right here mmm I remember there was a reason I wrote it again I see to the blind man so U is equal to Y to the negative five so we have to plug that in okay so it's gonna be e to the negative five x over Y to the five I'm bringing it down right because it's Y to the negative five so I'm bringing it down equal to X e to the negative 5x plus one-fifth e to the negative 5x plus C plus C unfortunately on the test I Direction say explicit solutions so we have to solve for y I really don't want to solve for y we can get the x y to the fifth and divide I could just say don't worry about it on this house and just leave it as explicit you want to do that instead oh let's do that then yeah so who cares so we're done right so so I'm a test if it says explicitly vit as implicit yeah very good Collin Collin Collin is right to make this look more beautiful we can divide by e to the negative 5x so it's gone and then plus I don't mess up in December and then you could bring it up make it positive but I'll just leave it there so no I wouldn't so yeah you don't have to solve for y don't worry about it so I'll remind me to tell the class on any of them one of them is fill in the blank so you do but it's it's easy it's not like this well I think it is and but the other ones just if it says explicit just don't worry about it it makes it easier to grade to a lot of people forget to solve for y I thought that they can't they just forget so kind of sucks to lose three or four points cuz you forget to solve ax Y so now you not to worry about it so any questions in the bernoulli took us 19 minutes yes that's it this or this one this one yeah okay Benson it's always it's always going to be e to the integral of P of X DX and your P of X is negative 5 it's always gonna be whatever's there so like let's say let's say Vincent I know it's hard to see over here but oh well let's say it was x squared y equals 2 in this case it would be e to the integral of x squared DX whatever is there you see it now yeah you sure good good all right so what do you want to do next we can do a home video homogeneous one just do what was the other thing you said homogeneous yeah yeah yeah like we said it ok yeah let's do it let's do a homo from the first test right homogenious from the first test yeah the second test doesn't yeah here we go this is two point five number three number three two point five number three I just picked one right this is random so y squared plus y X DX I think these are the hardest one yeah emphasizing the first task was your first test because of the hurricane was a take-home test right so so this test might seem really topic she study okay please do the review questions right usually in a regular classroom like the first test is pretty rough like people get bad grades okay but I shouldn't say that so there's two choices you can do x equals V Y in which case DX equals VD y plus y DV or or you can do y equals u X I'm having to dig deep here yes Matthew saw the white aha made it clear that that particulars for your AHA thank you yeah so how would you even know this is homogeneous that's a really good question yeah like that's a you that's a you that's a you that's a you it's just a bad V sorry it's a y equals u X I'm sorry my handwriting is bad I should do it like this dan was really like you Vee Matthew had a really good a question he says how do you know it's homogeneous this is really important for the test well the best way to know it's homogeneous is that you studied so much that when you're taking the funny be like oh that wasn't the review so like you're ready that will probably happen if you study but the real way to tell that without but that will probably happen to you yeah it's very you'll see but you look at the exponents right 2 1 plus 1 is 2 2 they all have the same exponent right and you can mix exponents that was the trick right so 2 2 2 so see how there's a 2 there and then one plus one is two they're both twos and there's a 2 here so it's homogeneous yeah so like this would be homogeneous as well because they're both threes right but if I do this game over right well I could do this as well that's also homogeneous because one plus two is three all right one plus two is three okay which one is easier x equals V y or y equals u X y equals u XY can you explain why yeah yeah so if you do this you have two things here and you put it here and yet the foil like ah so it's better to do this one so we're gonna do this one so solution Sol means solution so we have y equals u X I'm gonna write it again and so that means that dy equals y plus XD who focus on my tea use aha it's fine yeah I'll just make it confusing for me just get it right Kenny okay that's the best please try to get a hundred it makes it easier to grade so now you just plug everything in so you x squared so it's gonna be you squared x squared I just had to skip a step okay I'll do it I wasn't going to but since since Veronica said it so so you x times X I'm not gonna skip that step I feel guilty I just feel wrong doing it I won't do it right it's just you x squared right but it's you x times X I just I've been burned too many times on these minus x squared I think these are the hardest ones on the first test though homogenious like it's just really if you cuz if you if you mess up here it's like it would get it wrong right and then dy is here right here so it'll be u DX plus XD u and that's equal to zero it's a equal to zero yeah well good stuff in and then it becomes separable so basically after you do this it it becomes a separable differential equation so this will be u squared x squared plus u x squared DX all right and then minus x squared u DX all right I think I did that right yeah yeah let me finish I don't mess up X cube D u equals zero that times that is X cube yeah because Isaac said so and because the real reason is if you use this one there's two terms here and then when you put it there you have two things times two things so it's harder right so if we do this one it's just x squared times this it's easier to distribute but you give the same answer either way Matthew so yeah you get the same answer either way yeah and okay what we do now just distribute so this is yep yeah Levi doesn't the you I swear you yes yeah I'm gonna write it again I think you're right I'm just gonna one show one more step and I think you're writing it really fast mm-hmm I think you were right so I just rewrote it one more time and it does write these middle terms go away right that's a good sign sometimes stuff doesn't cancel right is that a bad sign no it's just sometimes stuff happens right so all right so now I'm gonna add this to the other side because this is gonna be a separable differential equation so we have u squared x squared DX equals x cubed do you do you so we're here so we're here so we only want to separate it right you want to get all the X's on one side with the DX and all the use on one side with with the D you right yep very good so divided by X cubed that will give you DX over X yep so pros do and then over here would be u over u squared yep which is U to the negative 2 if you really want to skip a step right because we do have to do what now integrate very good Jeff like a pro on fire I never see you this early you look different that's good no no smarter it's too early yeah the night class people don't wake up till 4 and I'm kidding though divided by X cubed so we got x squared over ah yep and then brought that over there so now we get to integrate so divided by X cubed x squared over X cube is 1 over X divided by u squared before I integrate this though what do you have to do to this one to integrate it bring it upstairs yeah bring it up this is gonna be U to the negative 2 D you write U to the negative 2 to you just not so bad like that's cuz you're helping me write like so this is gonna be Ellen I'm somebody of X yep equals U to the negative 1 well you know as long as he added on one side I had a girl my class once you'd put that she would do this every time she put teeth in it and stuff I don't know I thought it was funny she did all right she should have gotten tanked okay so this is this is Ellen she never did the homework equals this is negative one over you okay plus see I am so glad we don't have to solve for y I'm glad I made that rule that's great saves it saves us like 15 minutes of review time that's why I did it we're almost done right U is somewhere it's right here y equals UX right we're here so that means U is equal to Y over X right so 1 over u x over Y right it's a reciprocal plus C and then we don't have to worry about solving for y so that's it yeah do you see why because it's equal to UX yeah cuz it's flipped let me pause here no I'll stop let me stop for a minute and labor good it's good it's been 30 minutes and we've done to des it's about it's about right it's pretty fast that's faster than normal like when we do but when we did Bernoulli the first time it so it goes like half an hour to do the problem yeah but I'm going a little bit faster so sorry any questions so far on this one ok so I'm scared to do variation of parameters that takes so long do you want to do something that's easy for a change we can do a Laplace they do something easier first yeah be something easier then we'll do it applause I just want to do something easy let's do it so this is something that is from exam 2 and it's on your review questions and it might not be super clear what I'm what I mean so this is the one from old exam 2 so it's not the one you took it's the old version that has solutions and this is number 6 number six and the question is why double prime plus well you only have one of these so I just want to emphasize that you know you don't have to be a master just make sure you yeah so part a was fine why see you pause here a moment everybody catch out so you have some easy stuff on your test - no just just find my seat probably just one there might be two but that would be an accident so there's one there's one intentional Bernoulli there she needs to be one of each but again it's an it's intentional that you have one of each the auxiliary equation or characteristic good M squared plus one that comes from this right basically there's a second derivative so it's a two this is the zero if derivative so it's m to the zero so it goes away or if you remember that from before know we're finding this is called the so so we're finding YC so YC is the solution to the homogeneous equation so you pretend it's equal to zero do you remember and then you have to find the characteristic equation so you look at M Squared remember so you always throw that equal to zero yeah after after yeah so subtract one so you get this take the square root so you get M equals plus or minus what would go here in this case hi very good you'll remember this is it coming back let me go ahead and write the formula down because this is reviewed a so the formula was C 1 e to the Alpha X cosine beta X plus C 2 e to the Kenny Kenny Kenny so sine beta X [Music] all right so so that's the formula that's the formula so we have to identify alpha and beta in this case right so what would alpha B in this case zero very good and what would beta be in this case one everyone see why because it's because it's zero plus or minus one times I right so alpha 0 and beta is is 1 ok so so I'm overcomplicating it just because I'm worried like you don't get it so I mean this is it right the test is one Monday for you all in my a night class it's tonight so bad so e to the 0 is gonna be what what happens to that that's 1 so you just get cosine C 1 cosine X plus C 2 sine X that's it Part B so that's part a and then Part B we've defined the form of YP and that's it you don't have to actually solve the de that's it yeah because it's it's time issues right so form well yeah no you just defined the form so watch so remember this so this is so so first you do we do it in two steps so the initial and then the mod remember this yeah you only have one of these though okay so so the initial is based solely on this so we've got a sign he John Sheppard since we have a sine and a cosine X plus B cosine X plus CX Plus D sine X I'm running out of room I put my mod somewhere else so that's based solely off of this I'll pause here from it let everyone catch up so we have to find the variation of parameters yeah we can do one later today mm-hmm miss like things like 20 mins but we can do it okay so now you got to see if there's repetition here and here is there any repetition yes good B cosine X and D sine X repeat with these and so good Josh you multiply the whole thing by X so you have repetition between B cosine and design with these right with with these right with these so the multiply the whole thing by X so the final answer is the modified so byp and I'll go ahead and distribute the X you don't have to I like doing it it looks like this plus CX squared plus DX so CX squared plus DX sine X and that would be do you'll remember this from before from the past the initial comes from here okay and remember because the sine you thought sines and cosines because you have to have a linear because you have a thinking of a full linear in front of each and the repetition is the hardest part this was the hardest one right because the repetition is hard to see you have to multiply it out in your head to see the repetition you have like eight of these on your test or something it was ridiculous yeah Jeff the question will say find the form of YP yeah and then we'll say fine why see ya yeah it'll just ask you very shouldn't parameters it'll say use variation it's probably the last question on your test yeah yeah Matthew no not with me there's a video yeah on the playlist if you go to the playlist and you scroll down to where it says fine YP method of undetermined coefficients Part one and Part two and then finding the form of YP that does that that solves the worksheet the YP video it's the third one in the below that yeah from that yeah it's in the playlist there are no word you'll see it it's there I can email to you too if you know me yeah just I'll send you the video yeah I worked out the entire worksheet so hey all right lip loss right Laplace and people versus dual applause let's do alpha so full in here it's like two classes everyone's here from other classes so Laplace transform I can't yeah I can print I can print some copies yeah I should print some today this one I think this one's easy so this is seven point three number eleven y double prime plus I pick some of these to help you so this should be a good one this is the only Laplace on your whole test right just one de everyone did really well I'm a little lost transform toss most people so I figured that I should not make it super heavy but make it something right so no no just just one just yeah just just this yeah mmhmm yeah just touching on it right like one variation of parameters just taught it's a final so and the time right Michael this class only has 75 minutes okay first thing you do is take the Laplace of both sides right so solution so Laplace well I haven't done this since you've done it so plus two Laplace y prime applause of why the Philosopher's 0 which is 0 did we prove that in class one day right like on the side it was like a remark yeah yeah oh yeah yeah I remember I remember right yeah I think I did it on this board yeah please floor Gua yes from the ocean all right so I was thinking like that weapon that the guy in this ocean has trident yeah yeah so because it's the second derivative it starts with s squared pitch warp Y of s minus s Y of zero and it ends at one less derivative so it ends at Y prime that's how I know what to put a Y there because it ends up one less derivative it's always - yeah I had to really think they're from I got confused you didn't yeah you could yeah in my mind I was confused in my mind plus it's hard it's hard yeah it's a good test like you have to know de like huh that's the rule that's the part of the formula mm-hmm that's the formula mm-hmm this will be s pitchfork Y of s minus what your honor Y of 0 y at 0 at the rivet F at 0 so Y of 0 yeah it ends that one less derivative I like how you said that Josh he said zero with derivative the zero of derivative is the original function right I always think of that cuz it comes up in Taylor series anyways plus and then pitchfork why right pitch for y of s I love this stuff feel like I'm reviewing like a like this is good for me too like this is a lot of math like I wrote these I was eating breakfast isn't wanting I wrote these down like oh I should I should you know have something so you can take pictures of this too if you wanted for class these problems it but I wouldn't say that these are better than the others some of them are obviously but well plus of 0 is 0 yeah very good good Veronica yep so now we use our initial conditions so all of they're both ones how nice right so this is going to be s squared pitchfork Y of s minus s minus 1 plus 2 s pitchfork Y of s minus 2 right - - but that's a 1 it's a minus 2 so minus s minus 1 minus 2 Plus pitchfork Y of s and that's equal to 0-0 yep zero zero hey you made it good hey I'll put this on the internet like tomorrow or Thursday also oh it's tonight nevermind sorry yeah yeah I will for my morning class for you all yeah I will tomorrow or or Thursday I just have to download it it takes a while and then I have to like it's like six files okay what happens next we gotta solve for pitchfork why right so so I could factor it out so it'll be s squared plus 2's plus one right plus one plus one and then this is this is confusing I'm gonna skip some steps here this is negative s minus three so if you add it to the other side that's plus three very good all this problem is awesome Oh interesting okay I mean you could do that yeah mm-hmm right right you could do it that way yeah that works mm-hmm that certainly works good not divided by that it's off a pitchfork why this is like the easiest one in the homework but it's good s plus three and this is called a perfect square trinomial it's the name it has in math because if factors it's as possible boom squared that's plus one squared so we're here it's a nice problem this is a good test question we got to do the special I can't say anything is being reported so so so yeah we had to that special trick that's right so you really want it to be an s plus 1 so you just do whatever you want you put it there because we can do that plus 2 because you're trying to get 3 so you write down the 1 because I put you want so we're trying to get a 3 so we add 2 I loved that special trick it makes our lives so much easier so you write down what you want and then you fix it later you have a 3 so you just add a 2 it's beautiful isn't it Kenny these these cancels one over huh I don't know wait a minute what would you say it's just what yeah why oh oh yeah why don't you say y of S or pitchfork yeah this is pitchfork Y of s I don't know it's really bad ok so what's going on this is the Laplace of Y right that's what this is just for some understanding so the Laplace takes Y and sends it to this so the inverse Laplace takes this and sends it back to Y it's the inverse function so to finish all you have to do is take the inverse applause so Y is equal to the inverse Laplace of 1 over s plus 1 plus pull out that to pull out that 2 inverse Laplace oh you know what this is gonna seem hardcore but it's okay this is how we do it translation theorem it's just do the shift right because this one is in E you can do this shift here it's another formula sheet yeah David okay I thought I messed up ok this one you can do a shift too but it's on the formula sheet right and this one we do is shift because it's squared so this one's gonna be e to the negative T right cuz it's s minus a would be e to the a 2 so it's s minus negative 1 so it's e to the negative T plus two this is one factorial so it's going to be T right T and then we did a shift so it'll be e to the negative T yep and that would be the solution to the differential equation it's e to the negative T because of that this this one so this one has a factorial right so it's so it's one factorial so it gives you t to the one so let's check it out check it out say it was inverse Laplace three factorial as to the four asked to the s minus 6 that would be T cubed e to the 16 because this gives you the T cubed and this gives you the e to the 60 the shift of this you have you drop the you drop the e and becomes a shift the e becomes but that's not that shouldn't come up because it's just this yeah I put that I pulled the two out and then I wrote the one factor yes I can bring a couple of extra ones if you need it yeah sure yeah but this is if you could do this when you're pretty much ready like it's very light on Laplace I figure it's a final and does replace your lowest test so I had to justify that somehow so I had to put something with applause like series you only have one series question like an easy one like one just one just one seer it's an easy one everyone should get it right so we have 30 yeah I've times time uh okay did that hold on across the file so we did we did the Bernoulli wow we did the homo ha let's do it exact because it's like okay let's do all right well I crossed out the exact huh he's doing you guys do it exact first cuz that's shorter then we can then we can do variation if you want yeah a lot already like it's quite a bit like it was pretty good like I'm feeling better about the test two point four I saw this list I'm like there's no way we're gonna do all this we've done really well so this one is really evil this was on a final before and I remember a bunch of people got it wrong I'll tell you why dy equals zero so this is one of the review questions there's two point four number six from the homework yes it's a homo also and you see if you do it the home away it's really hard like it's super tough so we're gonna solve this is gonna be exact so we're gonna yeah but this home would yet M&N it is homogenious so Matthew said it is homogeneous so if you do it the homogeneous way you will get the answer but it's really hard okay like it's more work right so so what you want to do on the test is always check to see if it's exact so always do that first okay so this is our M and this is our and having deja vu because we kind of did this in calc 3 the other night right oh you can't count 3 so del M del Y right so I have to work that out it's the other variable it's how you memorize it right there's an X you put of Y yeah well we're gonna check so check if it's exact you call this M you call this N and then you compute del M del Y it's the other variable so it's gonna be 6 Y to the fifth okay now we have to compute another partial derivative that would be del n del X very good because it's the other variable and it's the same so it's exact so exact and it's not gonna be a tricky one I think you had one on your test with an e that required the product rule was that your test no it won't say and and as Matthew pointed out it's obviously homogeneous cuz look 6 6 1 plus 5 is 6 but don't do it that way right it's harder if you do it that way yeah I was just say salt and then you just cross your fingers and then just yeah yeah okay um so whenever it's exact just do it the cheesy way I'm gonna write it again up here it's gonna copy-paste it can't really copy paste we don't have those powers in real life you don't know you just have to yeah you got to be a mouth guard so no saw walked in late on the test that will say explicit solution don't worry about solving for y just implicit so k so okay integrate this with respect to X so it'll be X to the 7 yes exactly is easier to plus integrate with respect to X so integrate with respect to X so ant all right int wrt ah I feel like I should put a period there but I won't know integrate with respect to X integrate with respect to X integrate with respect to X it's really fun but I won't okay okay that's done okay and this is ax Y to the 6 plus an unknown function of what variable G of Y the other one very good that's equal to it's been a long time right it was like September right integrate this one with respect to Y very good oh yeah the six is cancel right so we just get X Y to the 6 it's over 6 plus h of X that's right the pause here so we integrated this yes yes the what the what way the new boy yeah it is yeah it is because yeah any questions yeah so easy trick so whenever there's an X just integrate with respect to X good whenever there's a y and can't respect to Y so your differential yep recap you're taking the test to see this question yes it's homogeneous don't do it that way check to see if it's exact find del M the other del Y del n del X are the same so it's exact it's always the other variable to check I don't know you'll find out if it's exact oh right if it's exact yeah so integrate with respect to X get this at an unknown function of Y integrate with respect to Y at an unknown function of X all right here's the key part now you write the answer down right left to right so you write that down because you see it yeah F X y equals C so I'm running down here the f of X Y so then it's this one ignore it oh we already wrote it down right so the answer is equal to what do you put here to remember see that's the answer yeah let me explain it again yeah yeah I'll explain it again so so you write this down you write this down right so you write it down write it down ignore it oh look you already wrote that down so you don't write it down again let me explain why the reason this happens is as follows this is equal to f of X Y this is equal to f of X Y so we basically have an equation that's saying f of X y equals f of X Y it's like saying 2 equals 2 duh we call that that's why it was different was a different question it wasn't an equation in that case it was a function no it's an equate now write it like this write it like this because if you remember this equals C came from the fact that it's equal to 0 when we had the total differential the DZ was 0 because C was a constant so it's a little bit different so it's a little bit different so I wouldn't you could you could put you could do this you could do f of XY equals because the final answer is of the form it has this form the final answer and that's because of the way everything was constructed remember the DZ and all that yeah no I don't know you don't have to specify it that's yeah well you know you'll know the Bernoulli right the Bernoulli will have that Y to a power right so you got that one you've got the holes down in exacts need to put equal to see yeah the end at the end you put equals see yeah it's good that's good we did the homo I don't think we need to do a linear right so we do we did Bernoulli exact homogeneous Laplace oh my god we've done everything variation you want to do it all rights do it I have two of them on here I'm not sure which one to do the hard one okay all right well no we didn't do linear and wait and do series stuff but but there are easy you not so so four six number five how many of these number one why double prime plus three y prime plus 2y equals one over nine plus e to the X does he write it down I'm gonna see if I can look I'm gonna look at the test really quick see if I see if I could find it yeah this one yeah this one's on your old test okay this one is on old exam too so you have the answer to this one except there's a 1 there and on the old exam but it's it's the same problem it's the same problem ok so it's been a while since you've done this so solution so the first thing you do is you solve the auxiliary yeah oh it'll say variation of parameters yeah thank you it will tell you by the way yeah yes by the longest thing on the example mm-hmm so first step is you write down the characteristic or auxiliary equation so it'd be M squared plus 3 M plus 2 equals 0 that's the first step that's the first step it's first step and then you have to solve this this usually factors I hope nicely let's see so it's M plus 1 M plus 2 you have M plus 2 very good M plus 1 M plus 2 and that's equal to 0 I'm alright a little bit small it's alright so we get negative 1 and negative 2 right negative 1 and negative 2 so we get M equals negative 1 M equals negative 2 I think will be a lot of good grades on this test now that we're reviewing it's really good I haven't reviewed I haven't reviewed for this final I think 80 years mm-hmm it's been like [Music] plus C 2 e to the very good good Matthew yep repetition mm-hmm which might not even come up I mean would you so so now we have to identify y1 and y2 so this is our y1 this is our y2 if you mix them up it's okay okay so y1 is e to the negative x y2 is e to the negative 2x ok and now you have to take the remember the primes the other derivatives very good so y1 prime this is B Pro here we're gonna get a negative because of the chain rule so B negative e to the negative x and then y2 prime will be negative - did you do it - oh that's ok all right so now you gotta find the wronskian right but to find W w1 w2 all of that stuff ok so W is the determinate and then it's just this so it's e to the negative X it's just the wronskian of y1 and y2 each of the negative 2x so it'll be here I'm really impressed with how much we accomplished today like I did not think we would do this much today like I'm like shocked we actually did a Bernoulli a homo I feel so I can't believe we did it like I was like there's no way we're gonna yeah Wow ok let me just stop talking let's do this so W so who can get this all wrong and not finish it so this times this negative 2 you add the exponents right so e to the negative 3 X and the minus this times this oh but it's already - so it's plus like that aha and I'm gonna put this in a box because this is a huge accomplishment in our lives all right so oh yeah this is Levi sorry yeah no it's yeah I mentioned in before in class really what I say that doesn't matter okay all right so so w-why don't wanna get that you have Levi w1w1 so for w1 you take the God good you replace it with this so replace the first column with zero and f of X so you do zero and then that I copy that down right I did okay and then you keep the second column okay mm-hmm always it's always this okay so let me go over that again so for w1 you replace the first column of the wronskian with zero and f of X right always every single time and I guess we'll do it so that's this times this so it's zero minus and then this times this Oh yuck each of negative two x and then nine plus e to the X I'm gonna write it again over here and I'm gonna put it in a little box cause it's really nice to box stuff that's always nice and people like box stuff you know makes it easier to grade always not some people have good handwriting do I have bad handwriting but W - that's the thing you do at the end of the year right yeah yeah you get that if you have a job they give you that so you keep the first keep the first one [Music] I'm not going to but you can and then you replace the second column with zero and f of X right the second column so so it'd be zero and f of X over here so zero this is the hardest one I think this is the hardest one in the homework besides number seven I didn't assign six and seven as review questions because they were just not fun this times this minus zero so it'd be e to the negative x over nine plus e to the X so we have w2 equals e to the negative x over nine plus e to the X and that would be W - don't you - there's a lot of points here this is one of the six questions on your test this variation so it's probably 20 30 points well very no I don't know about that you got this okay finally use now right you want in YouTube that's what we have to do right to use so you want yes yes the fun part I know just getting started gotta get some water so you one is W 1 over W so W is over here so we would I'll write it so it's going to be so W and W 1 are both negative so it's going to be positive okay so this is w 1 so I wrote down W 1 we're dividing by W so when you divide by W ya yes I multiply by e to the 3x which is the reciprocal is that okay can I do that they're both negative so they cancel good question good question so I did skip some steps here because we divided so we divided by W so when you divide by W you multiply by the reciprocal so I skip some steps here so good question no ask yes drill no we divided by this so you multiply by the reciprocal I skipped some steps sorry I should've just shown the step so so basically the negative and the negative cancel so it's positive when we divide by this we multiply by the reciprocal which is Kenny said it so I wrote it everyone see how that's okay I'm okay with that let's clean it up okay I'll show it I'll show it I'll show it so I'll show every step so it's so sorry Kenny so it's w-want so it's 9 plus e to the X and then divided by W so so it's times 1 over negative e to the negative 3 X looks ridiculous if I show it see it's worse because we're dividing by W because that's the formula that's more confusing now right so this is equal to it makes it harder for me so now we bring it upstairs that's good cuz you don't wanna mess up here cuz that that that's better and then this is equal to each of the X 9 plus e to the X DX and what was this call this is u1 this is a trivial integral right you let u be the bottom yeah yeah girls right it's you don't even need the absolute value cuz it's positive so yes I wanted to do that yeah so you can call it you the derivative is the numerator it gets an Allen with an absolute value but you don't need the absolute value because 9 plus e to the X is always positive anyways so you can drop the absolute value yeah it's rigged so I'm gonna come to this board now everything goes away if you put the plus e it'll get absorbed at the end later anyways so you could you could try it and you'll see it just all goes away now we got to find you to this is where it's gonna get harder but any questions on u1 we still have we still have a couple minutes so that's good u2 u2 u2 is w 2 over W so W 2 is way down there it's over here it's this one and so we're dividing by W so this time let's go ahead and skip that step we're gonna multiply by we're dividing by this so remote so it'll be negative so it'll be negative e to the 3x times W 2 which is e to the negative x over 9 plus e to the X DX right so it's this one divided by this one so when you divide by this you multiply by the reciprocal again the same thing happens mm-hmm right so this is the step that which I'm glad Matthew asked that I showed it's the same thing happens over there and you bring it up and it becomes positive so that was a good question in Matthew it's good yeah so these this is going to be u 2 equals negative e to the 2x and that's what people give up you can ix vehicle - yeah you could do it that way I'm gonna do a little bit different you could do it that way you could you could so you could make it use up and do it I'm gonna yeah yes oh it's divided though so see when you divide it does that and then when you bring it up it becomes positive it's good it's really full any questions makes sense all right all right hey you know what after the test it's Christmas right oh this is kind of fun okay there's a couple ways to do it Jeb had a really good idea his way but I'm gonna do it my way just because I don't want to mess up I don't have ten minutes I want to take a chance in ten minutes so my idea was to come to the side and do this so rewrite it like this right because e to the x times e to the X is e to the 2x and then you could do this you can you could split it up like this so you have e to the x over night remember this I mean you add nine and subtract nine number this trick so so basically now you put a plus nine here then you have to do a minus and I remember that that's what this is what legends are made of this is like mathematics for the soul so basically you're here right and then you add a nine that's what you want and then maybe at the end of class will do a Jeff's way and this is one minus nine over nine plus e to the X right everyone suggests want to pull it out just because e to the x times e to the X is e to the 2x [Music] that's all I did I'm sorry I can decode it so so we're here and that all you mean here I put it out here so we're here the same thing then we want a nine years we put it there and then we take it away okay so so now let's come back here so YouTube YouTube is going to be negative e to the X DX plus nine e to the X right to be plus now so plus nine right because it's plus 9 e to the x over 1 plus e to the X DX so that's this times this now you distribute when you're here so it's just a clever trick to take this and write it like this this is a really hard integral that's right that's right we do because we're pro but let me pause here because this is like the hardest part of the entire problem right this one makes its problem hard yeah you know what that's called that's called the mistake thank you where you for the same thing okay I'll get you after class oh yeah oh I'll give you the points yeah where you can just you can do it well this one's easy ah fail fail falling apart this one's easy this one this one you do a u sub though right for this one Veronica so if you wanted to do it I'll do it up here you would let u equal mine plus e to the X so D U is e to the X DX and then so this becomes this just becomes 9 ah 9 d u over u right and then it just works out so this is yeah we have it yeah so we just do that it's almost done almost done we have six minutes we're gonna need them we don't we need these six minutes Jeff had a really good idea by the way and it does work Jeff's idea was you do this okay and then you then you have to you have to think about it like this and you could you could do it with this substitution it will work right because your D u in this case is e to the X DX that takes care of that takes care of one of the ease and then you solve for the other e you get you - 19 you plug it back in so Jeff's method is a very legit method and it's very organized and it works so good Jeff I just want to emphasize that your way is really really good I was just scared to do it because we only have six minutes and huh so now what's the answer so now we go to find YP so I come back here so finally so now YP e we're getting there were getting there is U 1 y 1 plus u 2 y 2 fun times I think this is one of the ones that doesn't simplify oh my god it does nope we need every minute so that's why P that's a good test question okay so now you plug everything in so u1 is way over here it's this Ln 9 plus e to the X right so it's Ln 9 plus e to the X Y 1 I erased it with my hand you can still see it there's there's a faint survival piece of it they're right there e to the negative x u2 is that ridiculous thing we found earlier so parentheses negative e to the X plus 9 allen 900 plus e to the x times and then y2 is here it's each of the is it was it too much for negative 2x I think it was negative 2x it was negative 2x yeah so that's that's YP that's the YP this is another no not yet not yet not yet so now we have to find y so Y is you remember what it is it's my peoples YC before we do though I'm going to just distribute this I know it seems ridiculous but it's gonna be necessary in this problem so I'm gonna write it one more time so YP a line it's kind of sad like this is it you know de is over this times this you add the exponents right yeah you should because it's gonna simplify at the end Isaac how do I know that cuz I look the answer on 9e to the [Music] thing it's on the hold yeah you don't need it you can't have it it's fine to have it yeah yeah so finally that make seem long a dramatic finally finally we have arrived now we'll write that so so Y Z plus y P all right YC plus y P so I forgot what YC was called it was the one with the C's what thank you e to the negative 2x right trough thank you and then I'm gonna write all of it again I know it sounds ridiculous I'm gonna put the e in the front because it's just bothering me Allen 9 plus e to the X running fast there's one more step after this and we're done - e to the negative x plus 9 e to the negative 2x ln 9 plus e to the x almost dumb you just pause I can write faster with my marker than you can with your pen or pencil probably I read somewhere on the internet that people can write faster with markers is that true some teachers write really fast I've seen yeah there was a guy my office one day he was writing integral signs like this I couldn't even like it scared me how fast it was any questions any questions like no we're not done is one more step the problem will say simplify your answer as much as possible in bold so I think there's a common term here e to negative X yeah so in theory in theory you could take this one and take this one and you could write it like this see 1 minus 1 e 2 negative X see one is arbitrary it could be any number you subtract 1 you still don't know what it is it could be any number so just call it C 3 so so this is gonna be so combine these and call it C 3 e to the negative x ln 9 plus e to the x plus 9 e to the negative 2 x ln 9 plus e to the x it's not necessary though nothing will simplify these mean these here yeah that's writing but you could that's not supplying though simplifying is go ahead there's no there is no other way to the negative 2 X good question so how did you simplify good I'm glad you asked so basically you look at these and and this one and they're common terms right so what you can do in your mind is you can pull out the e to the negative X and write it like this see 1 minus 1 right cuz C 1 e to the negative X is there and the negative e to the negative X is there right so then not what you can do is you can just rename this because you can rename constants remember so this is C 3 e to the negative x so it's like if you had it's like if you had it's like if you had C 2 yes yes so it's like if you had C 2 e to the X e to the 3x plus 4 e to the 3x in this case just combine them and call it C 3 e to the 3 the same the same any questions on this stuff alright that's it that's it good luck good luck and fin this off

Channel: The Math Sorcerer

Views: 32,784

Keywords: differential equations, final, exam, differential, equations, math sorcerer, review, final exam review, maths, mathematics, differential equations final exam

Id: kIZpbeE_yTc

Length: 74min 13sec (4453 seconds)

Published: Fri Dec 06 2019