Equilibrium Solutions and Stability of Differential Equations (Differential Equations 36)

Hey welcome to another video I hope that you are ready to start wrapping up this stuff about population modeling because we only have about three more videos this one and two more and then we're going to move on to acceleration and some oscillation type problems so in order to wrap this up we're going to talk about how we didn't know when something is going to go to a number as far as a population or explode and so we're going to talk about something called equilibrium solutions we'll talk about critical points talking about stability funnels of spouts and then we'll move on and talk about harvesting and and stocking populations in another video so today's video I really want to get the point across to as as to how you would know when our population is going to explode or go to a number when you're not based on something like the logistic equation or an explosion extinction model when we're not we can't immediately see that and so we're gonna discuss that in this video today so let's get right to it let's look at a differential equation like the derivative of X with respect to time is X minus 4 now here's the premise to this question what if your initial let's say this is a population of X so maybe X is our population with respect to time what if your initial population or your initial x value was was 4 well what would what would that do because if you plug in a 4 right there 4 minus 4 is is 0 wait a second if 4 minus 4 is 0 that means that the way your X changes with respect to time would be 0 there would be zero change that's the the way that our population would be changing to make this respective time would you change our population of X or whatever X is would not be changing in in general if our derivative with respect to time is zero then obviously nothing would change here the rate of change is zero nothing's changing when that happens so when you have this value this point like 4 in this case or whatever is making your derivative 0 just like in calculus 1 we call that a critical do you remember that when you took a first derivative and you set it equal to zero and you said hey where this where the slope changes from positive to negative we have this horizontal tangent you go the slope is zero when the first derivative is equal to zero we have what's called a critical point and lots of important things happen to critical points same thing here if we have a model the way that some things change it with respect to time and if this equals zero at any point we gain what's called a critical point we have a critical point here so it's specifically x equals 4 is called a critical point oftentimes we'll also hear it called an equilibrium solution and why that is if you have an initial value of 4 so X sub 0 equals 4 and you plug that in here you can't possibly get a change from your population if we're gonna call some population or your X function if you want to call it that there will be no change it will stay the same forever never never never so if we start exactly at 4 there's gonna hold this equilibrium nothing will change from that you can't gain you can't lose it's going to be adding no change a 0 some sort of an equation so one time just to recap this one with our differential equations I'm trying to teach you when we will stay at the same level when we will explode when we will go to a carrying capacity or vice versa become extinct or come to a limiting population or when we're to stay the same I'm gonna come and talk about that to start there anytime your derivative is zero you have what's called a critical point we know that from calculus 1 here what that's gonna mean is that if you start at that critical point exactly then you would have a change in your population whatever your function is of zero your rate of change is zero it cannot change that solution of the critical point is oftentimes called an equilibrium solution because you are at equilibrium nothing's gaining nothing's losing so an equilibrium solution so how do we find them just like in calculus one's a very simple idea because your dx/dt is a derivative that's fantastic this is a derivative so our DX DT our derivative and X with respect to time is some function of X which that's what we have here if that's some function of X then where your derivative equals zero man see this where your derivative this is the derivative where your derivative is equal to zero it's going to give you these critical points so running through this if you want to find critical points all you got to do is set your function equal to zero because that function is the derivative here at critical points are found by setting your first derivative equal to zero so we're going to set our first derivative our function of X which is our differential equation we're going to set it equal to zero that's going to give us critical points so let's consider a critical point like x equals C a critical point we just kind of live through it but if X sub 0 your initial x is the critical point itself we call that an equilibrium solution nothing's going to change our derivative would be 0 you can't gain you can't lose we are horizontal right there so this would be an equilibrium solution yeah I suppose that's not all that exciting them it kind of makes sense right so if you if you get a derivative of X with respect time and and you know that's your derivative your first derivative of some function and you say hey I plug in some number that's gonna make it zero obviously we're not changed from that that's that's not a hugely surprising thing we're gonna just stay at the same level forever because the derivative the zero says we can't increase we can decrease or population stays the same or whatever your function Beck says stays the same forever the interesting things happen when your initial condition is above or below that critical point so let's talk what would happen if our initial condition is greater than C or less than C some interesting things are going to happen so wait wait a second if our initial condition is greater than C then really only two things could possibly happen now I'm not talking specific about this one differential equation but in general here's what could happen if we start here's our here's our critical point if we start above that then only two things could happen either we explode forever or we shrink to see if we start below that C we either gain 2 C or we become extinct or go to some lower critical point so we're either getting 2 or away from that critical point that should kind of makes sense the only two ways to go if you've got a critical point you start at it you're not going to change because your derivative would be zero it's an equilibrium solution be sort of above it we will either explode or we will shrink to see start below up you're either gonna gain 2 C or you're going to fall a real way from C if you want to kind of spout out of that or drop to the lower critical point or go extinct so if we have a initial condition greater than C we will either explode or shrink to that C what if it's less than C well we can either gain to C or we can become extinct if this is the only critical point these are the only outcomes so just to make sure we got that derivatives main rate of change obviously so if we set our first derivative equal to zero we have what are called critical points and if we have an initial condition that is that critical point nothing's gonna change slope would be zero function can't change you have an equilibrium solution the interesting thing happens when our initial condition is above or below that critical point if we're above we're either going to become extinct or string to that critical point if we're below it in the game that critical point will become extinct cliches how which ones which and what's beautiful about this because it's a first derivative because it represents the rate of change we can do the first derivative test or a sign analysis test or just look what the slope is doing and it will tell us exactly what's happening around those critical points so if our first derivative is this function we're talking on so this is a derivative I know it looks a little different than the first derivative test from calculus one because there's no prime but keep in mind this is the first derivative of some function X what we're gonna do is we're gonna simply put our critical points on a member line and to sort of a modified sign analysis test so well look at this engine just plug in a number plug in something greater than C and see what your function is doing see what your first derivative is doing and there's really four cases so sign nonces test first root test whatever it is there's really only four cases if we plug in a number greater than C and we get a negative then we plug in a number less than C to our first derivative right here and we get a positive what's happening is that our function would be decreasing if we have an initial condition greater than C it will be decreasing but but if we have a the initial condition lesson see our function would be increasing right there we have what is called a stable critical point what that means is that around that critical point in a neighborhood around that that you would have numbers whether you're less than or greater than you're going to ultimately go to that critical point that's called stability or that's a stable critical point so if we start above it it's going to drop to it start below it it's gonna gain to it this is what we call stable and how would it look I'm gonna draw that right here how would stable look if we have an initial condition less than our critical point so down here somewhere this says that we would be gaining so we'd say okay so you're testing this you're you look at your first derivative you found your critical points you have them on some sort of a sign analysis or a number line plug in a number less than that and you get a positive plug in a number less than that and you're getting positive wait a minute positive positive slope the function is increasing we would get something that looks like this well wait a minute if you plug in something higher you have a negative negative means negative slope means your function is decreasing we did something that looks like this so it's kind of messy but we have this kind of convergence of both of our sides here so we have a stable critical point if we have both whether we're positive or negative above or below that critical point we end up getting to it like an asymptote right there what you call that is that the funnel so everything's funneling to that number of see that's what's stable means that's what the picture would look like so what are the other options well what have you had this critical point and you plug in a number greater than that and you get a positive so that means that let's see if we plugging that were greater than a critical point we got a positive that that means increasing so positive so it means increasing that would be a function going away from that critical point gaining a way for remember you're above it and you're increasing or what if we plugged in a number less than our critical point to our first derivative and we got we got a negative well that would be decreasing that's a way from that critical point that situation right there is called unstable an unstable critical point is where your function tends to go away from that number it's like it's fleeing the scene all right this is a gravity this is like the stable means I'm gonna I'm gonna go right to it no matter what unstable means I'm gonna go away from it no matter what and you can see it right here if you have a if you got some number that's starting above your C and you find out that the first derivative is positive that means that you are increasing now you're gonna gain away from that number you're gonna get something that looks like this or below so if we have something that's made if we plug in a number less than or critical point in the first three maybe something negative then for every initial condition less than that critical we would have a negative slope they use so beats function is decreasing so I'm starting here and it's decreasing it's gonna look like this that is what we call an unstable critical point and a lot of people refer to that as a spout so the spout says I'm going away from it kind of going out from funnel means going into spout means going out from this particular critical point I hope it's making sense I hope that you're seeing the relationship hoping it very clear that what an equilibrium solution really means is that your slopes not changing if your slopes not change if your slope is zero if your slope is zero that means your function can't change if your slope is positive your function is increasing or so is negative your function is decreasing and then compared to our critical point how that interacts how your so interact around your critical point will tell you whether you have what's called a stable critical point or a funnel or unstable critical point or a spell going to or going away from that critical point and a neighborhood around it the other two conditions are called semi stable and let's take a look at it for a second what if you plug in a number greater than your scene you get a plus and you plug in a number less than you see any get any plus way to notice I mean listening so it would be positive so we would be increasing so be positive so we'd be increasing that seems really weird this is called a semi stable critical point because if we start below our C in this location below C we're going to have a positive so we will be increasing we're going to get to that C eventually so we're going to look like this so it looks something like that but if we're above that C we're going to be also increasing so it's kind of a 1/2 funnel for those values below and a half spouts for those values above this would be increasing still so we start above C we have a positive slope and the increase is still we get something that looks like this and so we have this below see you get to it above see you'd leave it that's called semi-stable likewise what if you plugged in a value above c and below so you get negative that means that we'd have a negative so things are function be decreasing in both of these cases well it's still semi-stable they look a little bit different for those values above C you'd have a negative slope that function would decrease but it would decrease until you get to C so it's gonna look like this so it's been a decrease until you get there but if you have something below C some initial condition below that critical point you're still decreasing the function is still dropping so you have this same sort of ideas where you have a a sort of us both a funnel and spout the same idea for those values initial conditions starting above C you'd be dropping to it for those below C you'd be going the extinction no I do want to say one thing about this this is for only one critical point so in the future when we have one or two or three critical points which we're going to see at some points you could have another critical point up here in which case this would gain and then get to it so around this one critical point we talked about whether we're leaving it or getting to it that's sort of the idea here so I'm only showing you one critical point but keep in mind I've got another one I could leave from one critical point and drop to another that's perfectly acceptable or but it is my lowest critical point if I leave it if I'm below it and I leave it if I spout out of it I'd be going to extinction if I have my highest critical point and I leave it I'm gonna be going to explosion I hope that makes sense I hope I've made it very clear have the interplay between critical and this funnel spout this stable unstable in semi-stable cases these are both called semi stable what I'm gonna do now is I'm gonna come back up here in other words you're gonna walk through this I'm gonna leave this on the board we're going to see what case this actually is so we talked about so far is that a first derivative is a slope when you set a first derivative equal to 0 you get critical points and now we're using this in terms of our population or our modelling situation to say what's the graph going to do around there if that's a differential equation what would the thunk should look like well it would depend on our initial condition and that's except that we're saying if our initial condition is 4 we have just this equilibrium solution if our initial condition is above my critical point I'm gonna be in one of several cases I'm either gonna drop to it or leave it leave it or drop to it depending on what my first proof is actually doing same thing for our initial conditions less than C so we're looking at this or saying it is a first derivative but since I don't know my parent function I'm kind of guessing at what it's gonna look like and it's based on our initial condition what it's actually going to do hope this makes sense I'm gonna come back up here I'll show you what this does okay so here we go number one we'd say this right here is a differential equation it means a first derivative so if X minus four equals zero then that's gonna say our slope is zero that's called a critical point so we're gonna say that x equals four is our critical point what we do after that is we do what's called a sign analysis test where you can hear a phase diagram or a first derivative test all the same thing we're gonna say our function our f of X which is X minus four right now is creating for us some sort of a phase diagram based on a critical point of four so we just take a forward you put on a number line and then we're gonna plug in just like every first true test ever I'm number greater than four and a number less than 4 and see what our sign is not giving a very simple case we'll go so much more advanced case in the next video but let's plug in a number like 5 five two my first derivative remember function is X minus four I'm going to get a positive now I don't care that it's going to be one I care that it's positive so this would be a positive slope it says that we aren't increasing now no get this the slope is positive slope is not increasing the slope is linear you can see that the slope is just linear X minus four here's what this says it says that because the slope is positive if I give it an initial condition greater than four my slope will be positive that family of curves would be a family of curves or all the soaps would be positive that means that the family of curves all of them would be an increasing function again because this represents the slope on finding critical points if I plug in a number greater than four it's that initial conditions more than four it's going to create for me positive slopes on a family and give me a family of curve that all has positive slopes and this this is coming from a family of curves positive slopes positive spokes means a family of curves each one of which would be increasing just like this so five is more than four five is more than a critical point we are getting a family curves positive slopes on that condition all of which would be increasing now plug in something like 3 or 0 then define to three minus four is negative one how care about the one so much I care that it's negative so think about that if I give myself an initial condition of three or something less than my critical point like three two one zero or whatever that means that I'm getting a family of curves here all of whom would have negative slopes that would be all decreasing all of these will be tracking away from that for think in your head is that stable or unstable we're going away from it it's it's spouting law it's going to get away from me this would be unstable in fact it's kind of look like this that's exactly what just said here you have a critical point see it's going to look like this at x equals four if we start above it we are exploding if we start below it we are extinct ting if you start at it we stay the same there's no other critical point so so get this like this is the only critical point right it's the only place where our slope could equal zero that's it so we started above that we are going to get a population explosion of the smallest population or a function explosion if we start below that we're going to get a population extinction or the function extinction is gonna go to zero this has to make sense right now I hope I hope you've seen it are you seeing the stability versus instability or unstability of this we're using the first derivative usually that I give so for saying positive means increasing negative means decreasing and showing where that is in relation to our critical point to determine whether we have a spout or a funnel stable versus unstable now can we go a little bit further and actually solve the differential equation well of course we can and we're gonna see why this is the way it is based on our differential equation so because we have some separable a separable equation here and we get our X's DX's and are constants DT and take this integral and exponent on both sides we've done this so many times ago and quiet kind of quick here and this whole thing is C if we replace that with C replace that with T do you remember this hope you remember as soon as you get that arbitrary constant especially we have L ends now we're going to plug in an initial condition you what do you mean an initial condition well let's just say that at X of 0 you had some sort of an initial condition that use the X of zero so that X of zero is happening at time zero okay so e to the zero power is 1 missus II would equal X sub zero minus four and if we plug that in so we have this X minus four equals x sub 0 minus 4 e to the T and just add our 4 and take a look at what this would do depending on what our initially I hope I didn't go too fast by the way this is just a separable differential equation we're saying our initial conditions X sub 0 so when we solve for our scene we have X sub 0 when T is 0 then C is X sub 0 minus 4 we've taken this plugged it in right there it's right there and we just add it forward but look at what happens look how this is a vertical shift of 4 that means everything happens around that for this right here is this number so it shifted up four units from the from x equals so from y equals zero so for e actually x equals zero from four units up now take a look at this right here look what would happen if X sub zero is more than four if X sub zero is more than four you have a positive number a positive number times the exponential man e to the T it's the same exponential you know so if this is more than four you have a positive exponential it's going to grow that's exactly what that says it says you start at four and then add a function to it that is growing that is this now what if this is less than four if this number your initial condition is less than four then something less than minus 4 would be something less than 4 months would be negative so you'd started four and say you so - oh my gosh wait a minute negative e to the T that that would look like this because it's a reflection across the white with zero line so x-axis so this is negative remember negative e to the T some sort of negative e to the T it's not a dropping like this it's a reflection of your exponential so instead of climbing you reflect it you'd be dropping that's what these say if your initial condition is less than 4 you have a negative exponential so go to 4 and then do a negative exponential that's why this looks the way that it does this is just a really nice way to see it so like when we go through our next set of examples in the next video what's gonna happen is I'm gonna do a lot of them just by finding my critical points and making these graphs it's very nice for a few of them we're gonna go all the way down and solve our differential equation and say do you see why this is do you see that that four is this critical point do you see that depending on depending on what your notional good condition is we get a positive exponential if it's above four a positive exponential that's below for a negative exponential it's going to spout away from that that's an unstable critical point this right now should be making a lot of sense to you I hope I've explained it well enough obviously I think it's really cool so it's it's very interesting to me that this stuff works so that you do a differential equation and model is exactly what we saw with our first derivative test so we're gonna come back we're gonna do one more example all the way through I'll explain the same things and then we'll be done all right so one more we're going to work our way through it I'm gonna try not to go too fast to make sure that we really really understand this but next time we're gonna do a lot of examples too so right now I just need the overview of what's gonna happen to be there we're gonna get stability or unstability and unstable critical point for each of these cases sometimes we can get semi-stable we'll see that next video how that looks but what needs to be there right now is what how the first derivative difference equation is the first derivative first-order difference equation how that relates to increase in decreasing depending on your initial condition that is what is key is is that you understand that because we don't have the actual function X we have the derivative of that function that when we go back to it we're getting a family of curves and that family of curves is dependent on what our initial condition actually is that arbitrary constant it's not so arbitrary if you have an initial condition is it because you solve for that and it says oh this is giving me a specific curve and where that specific curve is and how it behaves it depends on that initial condition of X that's a big deal and we're seeing that in the course of our our graphing here we're seeing how that initial condition drastically changes what happens so let's go through it we're noticing them get a first derivative differential equation straight to time and we have X 3 minus X and we say we'll wait a minute so we've got 3 minus X and that's equal to the first derivative or function then this itself that's a derivative and so we say we can find critical points all the time by setting our derivative equal to zero and solving so we have a critical point very easily of x equals three no now get this not always if you're above the critical point do you gain you can't say that and so I I wanted to make that clear when I first introduced this concept when first like ten fifteen minutes but I really can't do it by looking at X minus four because if we were above that we were gaining below that we were decreasing and you saw that on the first derivative but in general you can't make that statement you can't say every time I'm above a critical point I'm gaining that's not true because if I plug in a number like four I'm gonna get a negative so I'm decreasing that's why I said at the beginning that we're our critical point is if you start above it you can either increase or shrink to that could have a point start look you can either increase to it or if you go away from that critical point it all depends on how your first derivative behaves that's what we have to do the site analysis test we can't just arbitrarily say oh you're above it you're gonna climb forever that's not true right now we're sort of putting together the logarithm the logistic with X Mittman explosion extinction ideas in a more general term here so when we look at this they go yeah our critical points three let's go ahead take the time and do a Hearst root of test so our function is 3 minus X we've got a critical point of 3 all that needs to happen right now since we do not have our function itself we just need to understand what our slope of our function is doing this is the slope of our function so if we start with a value greater than 3 like for 3 minus 4 so our function here it's 3 minus X if we start with 4 we're going to get negative 1 we are understanding for every value above three without exception there's the only one critical point there's only one possible point where the sign of our function can change it's three every value above that three it's going to have a negative slope every single value initial condition above that is going to give you a family of curves with negative slopes every curve will have a negative slope every curve will be decreasing get them set against so important that there's only one critical point here there's only one number where our slope is going to change signs and that number is three every in every single initial condition greater than three is going to give you a curve remember this is a slope of a curve every initial condition greater than three is going to give you a curve that has a negative slope that means every single mission contains in greater than three it's going to give you a curve that is decreasing these are all going to be decreasing now do the same thing with a number less than your critical point like two or one or zero it doesn't really matter plug in 2 3 minus 2 is positive 1 so this is your slope all right this is your slope it's not your function it's telling you that your initial condition is the function you get is dependent on your initial condition so if I plug in a number less than 3 if I have an initial condition less than 3 what's in that saying is that that whole family'd curves every single initial condition lesson 3 is going to give me a function whose slope this is slope is positive whose function is increasing that right there is giving us what's called a stable critical point it is going to look just like this this is how this is going to look we're going to have some sort of a graph where the family of curves depends on what your initial condition is if we have an initial condition that's greater than 3 we are going to get functions that are decreasing if we get initial condition is less than this we're going to get functions that are increasing we're going to get what's called of the funnel things no matter where we start we're gonna funnel into something near that one number as time goes on do you see am i explained well enough the interplay between the first derivative test what critical points are from like I don't know how many semesters ago that was into what your family of curves has to be doing and the fact that it's based on your initial condition that's huge because in calc one you sort of sort of go backwards don't you you have a function and you take a derivative and then you say oh the function is doing this well here we're saying I have the derivative of the function what could the function you're doing well that's that's another question what that means is that I'm doing my my first derivative test but I'm using to model my function nothing that way around not to kind of gain my function not saying how the function looks but this is all the possible functions we would ever get and more than that it depends on where I start so if I'm starting above in this particular case I'm getting a family of curses all decreasing for every initial condition greater needs me for every initial condition lesson three I'm getting family curves all of whom are increasing and we'll put that together you have this final idea let's see how that plays out when we actually do our our solving of our differential equation so you see me do a few different techniques and give you a lot of different options over the last two or three videos on how you structure these I've done things like factored out negative when appropriate I've left them in there I've tried to give you a variety of ways to do it that way at least one of them are a little stick in this case I'm pulling out that negative and the reason why we do that it's not super important I think I've mentioned that many times is because when you start taking integrals I don't want you to take an integral of this and accidentally give me this because that's a problem you need a you sub and you get a negative so the reason why you'll see me do things like factor out a negative when it doesn't appear like you actually need to is just for that reason so in our case we have this Ln absolute value X minus 3t we've got negative T plus C sub 1 we have an Ln we're going to wait to plug in our initial condition until we take that both sides as an expert on e we have X minus 3 equals it's plus or minus e to the C sub 1 times e to the negative team does many many times this right here is our constant but we also know that constants are always our arbitrary constants are always found by an initial condition so what is our initial condition well we don't have a specific one remember what we're doing we're trying to be general we're trying to show the picture of a curse depending on initial conditions so when you get done with this you should be able to plug in any number and say hey for numbers greater than this number I'm going to explode or shrink to four numbers less than number I'm going to well may become extinct or gain to whatever that number is so we're going to use this initial condition to create a model of our general solution here whatever our initial condition happens to be let's plug it in when I plug in T equals zero I get X of zero that gives us one so our C is X of zero minus three now when we substitute that back in right here that's an X minus three equals x of 0 minus 3 e to the no eutteum I take just a sidestep for a moment and I'm going to tell you on some of the harder examples that we're gonna cover sometimes I will solve for C right now all of the oldest off to the side but I'll leave the seat in here and I'll solve this for X and then I'll plug the C back in because doing this on much more advanced examples can get really really really messy and so you'll see me okay plugins plugging your initial condition now leave your seat just before remember what that is solve for x2 then plug in your C later otherwise especially the fractions it gets it's pretty crazy so I'm going to show you some of those techniques in the next video as well now we're almost done we're quite there so we have noticed that's it that's a slope this is the slope of every single function of X that whole family of curves now we have determined that what that slope is the sine of that slope it depends on a critical point where this slope equals zero is where your signs could change things above zero or positive below zero negative so that critical point is is critical it tells us where so is positive and therefore increasing or negative and therefore decreasing and so what we determine because that's slope because that replicas are soaps of every general the general solution or every family of curves that would have this is a slope we're saying that everything above that value of three is going to give us a family of curves that is decreasing everything every value or sorry every initial condition lesson three is going to give us a family of curves who are increasing so we have a picture like this now let's make sure that what we saw that our differential equation actually models this so we had the slope of the function right here guys this is the function this is every possible function that would have this as a slope this is your general solution and it's based on whatever initial condition is now let's look that three hey that's a vertical shift that says that I'm gonna have some important stuff happening at that horizontal asymptote that equilibrium solution of x equals three that's important if my initial condition is three I get a horizontal line now what if my initial condition is more than three more than three well three something more than three months three would be positive but this is an exponential with a negative exponent that looks like this those are dropping remember that e to the negative T looks like that weightless that's what this looks like so if I have an initial condition greater than three I will get positive and positive decaying exponential just like it shows if I have something less than three I would get let's see again some lesson three minus three that would be a negative a negative decaying exponential so a decaying exponential flipped over so wait a minute if I take a negative so remember this is negative negatives reflect about the x-axis that would look like this this would be e to the negative T e to the it's negative e to the negative T either the negative T is a decaying exponential negative e to the negative T almost looks like a logarithm here but it's not that it's a reflected decaying expansion it says that you're taking this and flipping it over so it says says if you were going to be looking just like this for every value starting less than three that's what this means this would be a stable critical point this is called a funnel it's exactly what we get right here which I prefaced the last two I'm gonna do in another video so when we get there right now I really would encourage you to go back through and try these two on your own they take maybe five minutes once you understand them it's not that that hard my job today was to explain to you how the interplay the hell this is slope and therefore represents the slope of some parent function some general equation general solution and to show you that depending on your initial condition we're going to get a different type of curve something increases or decreases depending on what your initial condition is so I hope I've done a good job of that I hope that explained and you should understand what man sounds like a lot when I first introduced it you should understand now what an equilibrium solution is something that makes your slope zero it's also called a critical point you should understand that depending on whether we are greater than or less than that critical point determines whether we have an unstable leaving or stable coming to that critical point or both leaving and coming to or coming to and leading stable unstable and semi-stable cases so we've talked about equilibrium solutions we've talked about stable versus unstable we've talked about funnels and spouts and I hope that I've shown you the interplay on how the difference in equations actually show that but could you imagine could you imagine not actually doing this getting all the way down it's a what did be hard for you to see that it would be hard for me to see that this looks exactly like that without this first so this is really important your first derivative test is huge it shows you what's actually going on around those critical points then your differential equation says oh yeah now that I know what that picture is I'm able to kind of see it so I hope that makes sense because this is what we're gonna do for like the last three examples in the next video I'm just going to show you the pictures of them because we'll have a very good interpretation of what's going on so I'll see you for the next video when we talk about a few more examples and kind of refresh remember on stability and instability being stable and unstable spouts funnels and we'll talk about semi-stable have a great day I'll see you for that you

Channel: Professor Leonard

Views: 66,788

Keywords: Math, Leonard, Professor Leonard, Differential Equations, Equilibrium, Equilibrium Solution, Critical Point

Id: 7q33RFkMMpY

Length: 44min 27sec (2667 seconds)

Published: Mon Jun 17 2019