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Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | We get sine of t times the integral from 0 to t of this thing here. I could just take the 1 half out. So let me just, to keep things simple, so I'll put the 1 half out here, that's this 1 half. So 1 plus cosine of 2 tau, and all of that is d tau. That's this integral right there. And then we have this integral right he... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | So 1 plus cosine of 2 tau, and all of that is d tau. That's this integral right there. And then we have this integral right here. Minus cosine of t times the integral from, let me be very clear, this is tau is equal to 0. So this is tau is equal to 0. To tau is equal to t. And then this thing right here, I did some u s... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | Minus cosine of t times the integral from, let me be very clear, this is tau is equal to 0. So this is tau is equal to 0. To tau is equal to t. And then this thing right here, I did some u substitution. If u is equal to sine of t, then this becomes u. And we showed that du is equal to cosine, sorry, u is equal to sine ... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | If u is equal to sine of t, then this becomes u. And we showed that du is equal to cosine, sorry, u is equal to sine of tau. And then we showed that du is equal to cosine tau d tau. So this thing right here is equal to du. So it's u du. And let's see if we can do anything useful now. So this integral right here, the an... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | So this thing right here is equal to du. So it's u du. And let's see if we can do anything useful now. So this integral right here, the antiderivative of this is pretty straightforward. So what are we going to get? Let me write this outside part. So we have 1 half times the sine of t. And now let me take the antideriva... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | So this integral right here, the antiderivative of this is pretty straightforward. So what are we going to get? Let me write this outside part. So we have 1 half times the sine of t. And now let me take the antiderivative of this. This is going to be tau plus, the antiderivative of this is going to be 1 half sine of 2 ... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | So we have 1 half times the sine of t. And now let me take the antiderivative of this. This is going to be tau plus, the antiderivative of this is going to be 1 half sine of 2 tau. I mean, we could have done a u substitution. We could have said u is equal to 2 tau and all of that. But I think you could do that from rec... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | We could have said u is equal to 2 tau and all of that. But I think you could do that from recognition. And if you don't believe me, you just have to take the derivative of this. 1 half sine of 2 tau is the derivative of this. You multiply, you take the derivative of the inside. So that's 2. So the 2 and the 1 half can... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | 1 half sine of 2 tau is the derivative of this. You multiply, you take the derivative of the inside. So that's 2. So the 2 and the 1 half cancel out. And then the derivative of the outside. So cosine of 2 tau. And you're going to evaluate that from 0 to t. And then we have minus cosine of t. And then we're going to hav... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | So the 2 and the 1 half cancel out. And then the derivative of the outside. So cosine of 2 tau. And you're going to evaluate that from 0 to t. And then we have minus cosine of t. And then we're going to have, we take the antiderivative of this. Let me do this on the side. So the integral of u du, that's trivially easy.... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | And you're going to evaluate that from 0 to t. And then we have minus cosine of t. And then we're going to have, we take the antiderivative of this. Let me do this on the side. So the integral of u du, that's trivially easy. That's 1 half u squared. Now, that's 1 half u squared. But what was u to begin with? It was sin... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | That's 1 half u squared. Now, that's 1 half u squared. But what was u to begin with? It was sine of tau. So the antiderivative of this thing right here is 1 half u squared, but u is sine of tau. So it's going to be 1 half u, which is sine of tau squared. And we're going to evaluate that from 0 to t. And we didn't even ... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | It was sine of tau. So the antiderivative of this thing right here is 1 half u squared, but u is sine of tau. So it's going to be 1 half u, which is sine of tau squared. And we're going to evaluate that from 0 to t. And we didn't even have to do all this u substitution the way I often do it in my head. I see the sine o... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | And we're going to evaluate that from 0 to t. And we didn't even have to do all this u substitution the way I often do it in my head. I see the sine of tau, cosine of tau. If I have a function and I have its derivative, I can treat that function just like as if I had an x there. So it would be sine squared of tau over ... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | So it would be sine squared of tau over 2, which is exactly what we have there. So it looks like we're in the home stretch. We're taking the convolution of sine of t with cosine of t. And so we get 1 half sine of t. Now, if I evaluate this thing at t, what do I get? I get t plus 1 half sine of 2t. That's when I evaluat... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | I get t plus 1 half sine of 2t. That's when I evaluated at t. And then from that, I need to subtract and evaluate it at 0. So minus 0 minus 1 half sine of 2 times 0, which is just sine of 0. So this part right here, this whole thing right there, what does that simplify to? Well, this is 0. Sine of 0 is 0, so this is al... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | So this part right here, this whole thing right there, what does that simplify to? Well, this is 0. Sine of 0 is 0, so this is all 0. So this first integral right there simplifies to 1 half sine of t times t plus 1 half sine of 2t. Now, what does this one simplify to over here? Well, this one over here, you have minus ... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | So this first integral right there simplifies to 1 half sine of t times t plus 1 half sine of 2t. Now, what does this one simplify to over here? Well, this one over here, you have minus cosine of t. And we're going to evaluate this whole thing at t. So you get 1 half sine squared of t minus 1 half sine of 0 squared, wh... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | So that's just minus 0. So so far, everything that we have written simplifies to, let me multiply it all out. So I have 1 half t sine of t, right? I'm just multiplying those out. Plus 1 fourth sine of 2t. And then I have over here, I have minus 1 half sine squared t times cosine of t, right? I just took the minus cosin... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | I'm just multiplying those out. Plus 1 fourth sine of 2t. And then I have over here, I have minus 1 half sine squared t times cosine of t, right? I just took the minus cosine t and multiplied it through here, and I got that. Now, this is a valid answer, but I suspect that we can simplify this more, maybe using some mor... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | I just took the minus cosine t and multiplied it through here, and I got that. Now, this is a valid answer, but I suspect that we can simplify this more, maybe using some more trigonometric identities. And this guy right there looks ripe to simplify. We know that the sine of 2t, another trig identity you'll find on the... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | We know that the sine of 2t, another trig identity you'll find on the inside cover of any of your books, is equal to the sine, is 2 times the sine of t times the cosine of t. So if you substitute that there, what does our whole expression equal? You get this first term. Let me scroll down a little bit. You get 1 half t... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | You get 1 half t times the sine of t plus 1 fourth sine of t times this thing in here. So times 2 sine of t cosine of t. Just a trig identity. Nothing more than that. And then finally I have this minus 1 half sine squared t cosine of t. No one ever said this was going to be easy, but hopefully it's instructive on some ... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | And then finally I have this minus 1 half sine squared t cosine of t. No one ever said this was going to be easy, but hopefully it's instructive on some level. At least it shows you that you didn't memorize your trig identities for nothing. So what is this simple? So let me rewrite the whole thing. Or let me just rewri... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | So let me rewrite the whole thing. Or let me just rewrite this part. So this is equal to 1 fourth. Now I have, well let me see, 1 fourth times 2. So it's really 1 fourth times 2 is 1 half. And then sine squared of t. This sine times this sine, sine squared of t. Cosine of t. And then this one over here is minus 1 half ... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | Now I have, well let me see, 1 fourth times 2. So it's really 1 fourth times 2 is 1 half. And then sine squared of t. This sine times this sine, sine squared of t. Cosine of t. And then this one over here is minus 1 half sine squared of t cosine of t. And luckily for us, or lucky for us, these cancel out. And of course... |
Introduction to the convolution Laplace transform Differential Equations Khan Academy.mp3 | And of course we had this guy out in the front. We had this 1 half t sine t out in front. Now this guy cancels with this guy. And all we're left with through this whole hairy problem, and this is pretty satisfying, is 1 half t sine of t. So we just showed you that the convolution, if I define, let me write our result. ... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | And it doesn't happen exactly like this, but it can be approximated by the unit step function. Similarly, sometimes you have nothing happening, you have nothing happening for a long period of time, nothing happens for a long period of time, and then whack, something hits you really hard and then goes away, and then not... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | So wouldn't it be neat if we had some type of function that could model this type of behavior? And our ideal function, what would happen is is that nothing happens until we get to some point, and then bam, it would get infinitely strong, but maybe it has a finite area, and then it would go back to zero and then go like... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | And let's say that the area under this, I mean, it becomes very, to call this a function is actually kind of pushing it, and this is beyond the math of this video, but we'll call it a function in this video. But what we can, you say, well, how do you even, what good is this function for? How can you even manipulate it?... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | And I'm gonna make one more definition of this function. So what I just do here, let's say we call this function, we represent it by the delta, and that's what we do represent this function by. It's called the Dirac delta function. And we'll just informally say, look, when it's in infinity, it pops up to infinity when ... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | And we'll just informally say, look, when it's in infinity, it pops up to infinity when x is equal to zero, and it's zero everywhere else when x is not equal to zero. And you say, how do I deal with that? How do I take the integral of that? And to help you with that, I'm gonna make a definition. I'm gonna tell you what... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | And to help you with that, I'm gonna make a definition. I'm gonna tell you what the integral of this is. This is part of the definition of the function. I'm gonna tell you that if I were to take the integral of this function from minus infinity to infinity, so essentially over the entire real number line, if I take the... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | I'm gonna tell you that if I were to take the integral of this function from minus infinity to infinity, so essentially over the entire real number line, if I take the integral of this function, I'm defining it. I'm defining it to be equal to one. I'm defining this. Now, you might say, Sal, you didn't prove it to me. N... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | Now, you might say, Sal, you didn't prove it to me. No, I'm defining it. I'm telling you that this delta of x is a function such that its integral is one. So it has this infinitely narrow base that goes infinitely high, and the area under this, I'm telling you, is of area one. And you're like, hey, Sal, that's a crazy ... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | So it has this infinitely narrow base that goes infinitely high, and the area under this, I'm telling you, is of area one. And you're like, hey, Sal, that's a crazy function. I want a little bit better understanding of how someone can construct a function like this. So let's see if we can satisfy that a little bit more... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | So let's see if we can satisfy that a little bit more. But then once that's satisfied, then we're gonna start taking the Laplace transform of this, and then we'll start manipulating it and whatnot. Let's see if we can complete this delta right here. Let's say that I constructed another function. Let's call it d sub tau... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | Let's say that I constructed another function. Let's call it d sub tau. And this is all just to satisfy this craving for maybe a better intuition for how this Dirac delta function can be constructed. And let's say my d sub tau of, well, let me put it in, it's a function of t, because I wanna, everything we're doing in ... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | And let's say my d sub tau of, well, let me put it in, it's a function of t, because I wanna, everything we're doing in Laplace transform world, everything's been a function of t. So let's say that it equals, let's say that it equals one over two tau, and you'll see why I'm picking these numbers the way I am. One over ... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | Everywhere else. So this type of, this equation, this is more reasonable. This will actually look like a combination of unit step functions, and we can actually define it as a combination of unit step functions. So if I draw, that's my x-axis. That's my x-axis. And then if I put my y-axis right here, that's my y-axis, ... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | So if I draw, that's my x-axis. That's my x-axis. And then if I put my y-axis right here, that's my y-axis, y, and this is, sorry, this is the t-axis. To get out of that habit, this is a t-axis, and I mean, we could call it, well, we could call it the y-axis or the f of t-axis or whatever we wanna call it. That's the d... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | To get out of that habit, this is a t-axis, and I mean, we could call it, well, we could call it the y-axis or the f of t-axis or whatever we wanna call it. That's the dependent variable. So what's going to happen here? It's gonna be zero everywhere until we get to minus t, and then at minus t, we're gonna jump up to s... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | It's gonna be zero everywhere until we get to minus t, and then at minus t, we're gonna jump up to some level. So let me put that point here. So this is minus tau, and this is plus tau. Right? Minus tau and plus tau. So it's gonna be zero everywhere, and then at minus tau, we jump to this level, and then we stay consta... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | Right? Minus tau and plus tau. So it's gonna be zero everywhere, and then at minus tau, we jump to this level, and then we stay constant at that level until we get to plus tau. And that level, I'm saying, is one over two tau. So this point right here on the dependent axis, this is one over two tau. So why did I constru... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | And that level, I'm saying, is one over two tau. So this point right here on the dependent axis, this is one over two tau. So why did I construct this function this way? Well, let's think about it. What happens if I take the integral, let me write a nicer integral sign. If I took the integral from minus infinity to inf... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | Well, let's think about it. What happens if I take the integral, let me write a nicer integral sign. If I took the integral from minus infinity to infinity of d sub tau of t, dt. What is this going to be equal to? Well, if you just, I mean, if the integral's just the area under this curve, this is a pretty straightforw... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | What is this going to be equal to? Well, if you just, I mean, if the integral's just the area under this curve, this is a pretty straightforward thing to calculate. Right? You just look at this, and you say, well, this is, first of all, it's zero everywhere else. It's zero everywhere else, and it's only the area right ... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | You just look at this, and you say, well, this is, first of all, it's zero everywhere else. It's zero everywhere else, and it's only the area right here. I mean, I could write this, I could rewrite this integral as the integral from minus tau to tau. We don't care, infinity, minus infinity or positive infinity, because... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | We don't care, infinity, minus infinity or positive infinity, because there's no area under any of those points of one over two tau d tau. We could write it, dt, sorry, one over two tau dt. So we could write it this way too, right? Because we can just take the boundaries from here to here, because we get nothing, the w... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | Because we can just take the boundaries from here to here, because we get nothing, the whole other, whether t goes to positive infinity or minus infinity, and then over that boundary, the function is a constant, one over two tau. So we could just take this integral, and either way we evaluate it, we don't even have to ... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | What's the base? The base is two tau, right? The base is two tau. You have one tau here and then another tau there. So it's equal to two tau times your height. And your height, I just said, is one over two tau. One over two tau. |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | You have one tau here and then another tau there. So it's equal to two tau times your height. And your height, I just said, is one over two tau. One over two tau. So your area for this function, or for this integral, is going to be one. You could evaluate this, you could get this is going to be equal to, you take the a... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | One over two tau. So your area for this function, or for this integral, is going to be one. You could evaluate this, you could get this is going to be equal to, you take the antiderivative of one over two tau, you get, I'll do this just to satiate your curiosity, t over two tau, and you have to evaluate this from minus... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | You put tau in there, you get tau over two tau, and then minus minus tau over two tau. And then you get tau plus tau over two tau, that's two tau over two tau, which is equal to one. Maybe I'm beating a dead horse. I think you're satisfied that the area under this is going to be one, regardless of what tau was. I kept ... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | I think you're satisfied that the area under this is going to be one, regardless of what tau was. I kept this abstract. Now, if I take smaller and smaller values of tau, what's going to happen? If my new tau is going to be, let's say here, let's say my new tau is going to be there. I'm just going to pick my new tau the... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | If my new tau is going to be, let's say here, let's say my new tau is going to be there. I'm just going to pick my new tau there. Then my one over two tau, tau is now a smaller number, so when it's in the denominator, my one over two tau is going to be something like this. It's going to be something like this. I'm just... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | It's going to be something like this. I'm just saying, if I pick smaller and smaller tau, so if I pick an even smaller tau than that, then my height is going to have to be higher. My one over two tau is going to have to even be higher than that. I think you see where I'm going with this. What happens as the limit as ta... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | I think you see where I'm going with this. What happens as the limit as tau approaches zero? What is the limit as tau approaches zero of my little d sub tau function? What's the limit of this? Well, these things are going to go infinitely close to zero, but this is the limit. They're never going to be quite at zero. Yo... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | What's the limit of this? Well, these things are going to go infinitely close to zero, but this is the limit. They're never going to be quite at zero. Your height here is going to go infinitely high, but the whole time, I said no matter what my tau is, because it was defined in very arbitrary terms, my area is always g... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | Your height here is going to go infinitely high, but the whole time, I said no matter what my tau is, because it was defined in very arbitrary terms, my area is always going to be one. So you're going to end up with your Dirac delta function. Let me write it, I was going to write it in x again. Your Dirac delta functio... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | Your Dirac delta function is a function of t. Because of this, if you ask, what's the limit as tau approaches zero of the integral from minus infinity to infinity of d sub tau of t, dt? Well, this should still be one, right? Because this thing right here, this evaluates to one. So as you take the limit as tau approache... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | So as you take the limit as tau approaches zero, and I'm being very generous with my definitions of limits and whatnot. I'm not being very rigorous, but I think you can kind of understand the intuition where I'm going. This is going to be equal to one. And so by the same, I guess, intuitive argument, you could say that... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | And so by the same, I guess, intuitive argument, you could say that the limit as from minus infinity to infinity of our Dirac delta function of t, dt, is also going to be one. And likewise, Dirac delta function, I mean, this thing pops up to infinity at t is equal to zero, right? This thing, if I were to draw my x-axis... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | And you normally draw it like that, and you normally draw it so it goes up to one to kind of depict its area, but you actually put an arrow there, and so this is your Dirac delta function. But what happens if you want to shift it? What happens if you want to shift it? What would, how would I represent, how would I repr... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | What would, how would I represent, how would I represent my, let's say I want to do t minus three. What would the graph of this be? Well, this would just be shifting it to the right by three. For example, when t equals three, this will become the Dirac delta of zero. So this graph, this graph will just look like this. ... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | For example, when t equals three, this will become the Dirac delta of zero. So this graph, this graph will just look like this. This will be my x-axis. Let's say that this is my y-axis. Let me just make that one. And let me just draw some points here. So it's one, two, three. |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | Let's say that this is my y-axis. Let me just make that one. And let me just draw some points here. So it's one, two, three. That's t is equal to three. Did I say that was x-axis? That's my t-axis. |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | So it's one, two, three. That's t is equal to three. Did I say that was x-axis? That's my t-axis. So this is t equal to three. And what I'm gonna do here, it's the Dirac delta function, it's gonna be zero everywhere. Everywhere. |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | That's my t-axis. So this is t equal to three. And what I'm gonna do here, it's the Dirac delta function, it's gonna be zero everywhere. Everywhere. Zero everywhere, but then right at three, it goes infinitely high. And obviously we don't have enough paper to draw an infinitely high spike right there. So what we do is ... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | Everywhere. Zero everywhere, but then right at three, it goes infinitely high. And obviously we don't have enough paper to draw an infinitely high spike right there. So what we do is we draw an arrow. We draw an arrow there. And the arrow, we usually draw the magnitude of the area under that spike. So we do it like thi... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | So what we do is we draw an arrow. We draw an arrow there. And the arrow, we usually draw the magnitude of the area under that spike. So we do it like this. And let me be clear, this is not telling me that the function just goes to one and then spikes back down. This tells me that the area under the function is equal t... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | So we do it like this. And let me be clear, this is not telling me that the function just goes to one and then spikes back down. This tells me that the area under the function is equal to one. This spike would have to be infinitely high to have any area, considering it has no, an infinitely small base. So the area unde... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | This spike would have to be infinitely high to have any area, considering it has no, an infinitely small base. So the area under this impulse function or under this Dirac delta function. Now this one right here is t minus three. But your area under this is still going to be one. And that's why I made the arrow go to on... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | But your area under this is still going to be one. And that's why I made the arrow go to one. If I were to, let's say I wanted to graph, let me do it in another color. Let's say I wanted to graph two times the Dirac delta of t minus two. How would I graph this? Well, I would go to t minus two. When t is equal to two, y... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | Let's say I wanted to graph two times the Dirac delta of t minus two. How would I graph this? Well, I would go to t minus two. When t is equal to two, you get the Dirac delta of zero. So that's where you'd have your spike. And we're multiplying it by two. So you would do a spike twice as high like this. |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | When t is equal to two, you get the Dirac delta of zero. So that's where you'd have your spike. And we're multiplying it by two. So you would do a spike twice as high like this. Now both of these go to infinity, but this goes twice as high to infinity. I know this is all being a little ridiculous now, but the idea here... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | So you would do a spike twice as high like this. Now both of these go to infinity, but this goes twice as high to infinity. I know this is all being a little ridiculous now, but the idea here is that the area under this curve should be twice the area under this curve. And that's why we make the arrow go to two, to say ... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | And that's why we make the arrow go to two, to say that the area under this arrow is two. The spike would have to go infinitely high. So this is all a little abstract, but this is a useful way to model things that are kind of very jarring. That all of a sudden, obviously nothing actually behaves like this, but there ar... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | That all of a sudden, obviously nothing actually behaves like this, but there are a lot of phenomena in physics or the real world that kind of have this spiky behavior. Instead of trying to say, oh, what does that spike exactly look like? We say, hey, that's a Dirac delta function. And we'll dictate its impulse by some... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | And we'll dictate its impulse by something like this. And just to give you a little bit of motivation behind this, and I was gonna go here in the last video, but then I kind of decided not to. But I'm just gonna show it, because I've been doing a lot of differential equations and I've been giving you no motivation for ... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | But you can imagine, if I have just a, say, a wall, and then I have a spring attached to some mass, some mass right there. And let's say that this is the natural state of the spring. So the spring would wanna be here. So it's been stretched a distance y from its kind of natural, where it wants to go. And let's say I ha... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | So it's been stretched a distance y from its kind of natural, where it wants to go. And let's say I have some external force, external force right here. Let's say I have some external force right here on the spring. And of course, let's say it's ice on ice, there's no friction in all of this. I just wanna show you that... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | And of course, let's say it's ice on ice, there's no friction in all of this. I just wanna show you that I can represent this behavior of this system with a differential equation. And actually things like the unit step function and the Dirac delta function actually start to become useful in this type of an environment.... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | So we know that, we know that F is equal to mass times acceleration. That's the basic physics right there. Now, what are all of the forces on this mass right here? Well, you have this force right here, and we'll say this in the positive rightward direction. So it's that force. Then you have a minus force from the sprin... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | Well, you have this force right here, and we'll say this in the positive rightward direction. So it's that force. Then you have a minus force from the spring, right? The force from the spring is Hooke's law. It's proportional to how far it's been stretched from its kind of natural point. So its force in that direction ... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | The force from the spring is Hooke's law. It's proportional to how far it's been stretched from its kind of natural point. So its force in that direction is going to be ky, or you could call it minus ky, because it's going in the opposite direction of what we've already said is a positive direction. So the net forces o... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | So the net forces on this is F minus ky, and that's equal to the mass of our, it's equal to the mass of our object times its acceleration. Now, what's its acceleration? If its position is y, so if y is equal to position, if we take the derivative of y with respect to t, y prime, which we could also say dy dt, this is g... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | And then if we take the derivative of that, y prime prime, which is equal to d squared y with respect to dt squared, this is equal to acceleration. Acceleration. So instead of writing a, we could write y prime prime. Y prime prime. And so if we just put this on the other side of the equation, what do we get? We get the... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | Y prime prime. And so if we just put this on the other side of the equation, what do we get? We get the force, this force, not just this force, this is just F equals ma, but this force is equal to the mass of our object times the acceleration of the object plus whatever the spring constant is for this spring, plus k ti... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | So if you had no outside force, if this was zero, you'd have a homogeneous differential equation, and in that case, the spring would just start moving on its own. But now this F, all of a sudden, kind of the non-homogeneous term, it's what the outside force you're applying to this mass. So if this outside force was som... |
Dirac delta function Laplace transform Differential Equations Khan Academy.mp3 | This is saying that at time is equal to two seconds, we're just gonna jar this thing to the right, and it's going to have an, and I'll talk more about it, it's gonna have an impulse of two. Its force times time is going to be, or its impulse is gonna have a one. So it's, and I don't wanna get too much into the physics ... |
Logistic differential equation intuition First order differential equations Khan Academy.mp3 | And when you actually try to solve this differential equation you try to find an n of t that satisfies this, we found that an exponential would work. An exponential satisfies this differential equation. And it would look like this visually. It would look like this visually, where you're starting at a population of n na... |
Logistic differential equation intuition First order differential equations Khan Academy.mp3 | It would look like this visually, where you're starting at a population of n naught. This is the time axis, this is the population axis. And as time increases, population increases exponentially. Now we said there's an issue there. What if Thomas Malthus is right? That the environment can't support, let's say that the ... |
Logistic differential equation intuition First order differential equations Khan Academy.mp3 | Now we said there's an issue there. What if Thomas Malthus is right? That the environment can't support, let's say that the environment can't support, let me do this in a new color, let's say that the environment really can't support more than k, more than a population, more than a population of k. Then clearly the pop... |
Logistic differential equation intuition First order differential equations Khan Academy.mp3 | They're not going to be able to have food or water or resources or whatever it might be. They might generate too much pollution. Who knows what it might be. And so this first stab at modeling population doesn't quite do the trick, especially if you are kind of in Malthus' camp. And that's where P.F., and once again, I'... |
Logistic differential equation intuition First order differential equations Khan Academy.mp3 | And so this first stab at modeling population doesn't quite do the trick, especially if you are kind of in Malthus' camp. And that's where P.F., and once again, I'm sure I'm mispronouncing the name, Verhulst, who is going to come into the picture, because he read Malthus' work and said, well, yeah, I think I can do a p... |
Logistic differential equation intuition First order differential equations Khan Academy.mp3 | Let's set up another differential equation. And now let's say, okay, instead of, you know, if N is substantially smaller than what the environment can support, then yeah, that makes sense to have exponential growth. That makes sense to have exponential growth. But maybe we can dampen this, or maybe we can bring this gr... |
Logistic differential equation intuition First order differential equations Khan Academy.mp3 | But maybe we can dampen this, or maybe we can bring this growth to zero as N approaches, as N approaches K. And so how can we actually modify this? Maybe we can multiply it by something that for when N is small, when N is much smaller than K, this term right over here is going to be close to one. And when N is close to... |
Logistic differential equation intuition First order differential equations Khan Academy.mp3 | So let me write that. When N, so these are our goals for this term right over here. When N is much smaller, so much smaller, much smaller than K, so now the population is not constrained at all. People can have babies, and those babies can be fed, and then they can have babies, et cetera, et cetera. Then this thing sho... |
Logistic differential equation intuition First order differential equations Khan Academy.mp3 | People can have babies, and those babies can be fed, and then they can have babies, et cetera, et cetera. Then this thing should be close to one. And so then you have essentially our old model. But then as N approaches K, when N, as N approaches K, then this thing should approach, then this term, this term or this expr... |
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