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Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So if I define my function here, will this work? Well, this unit step function, when we pass pi, is only going to be equal to 1. But we want this thing to equal to 0, so it has to be 2 minus 2. So I'll have to put a 2 here. And this should work. When we go from any value below pi, when t is less than pi here, this beco... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So I'll have to put a 2 here. And this should work. When we go from any value below pi, when t is less than pi here, this becomes 0. So our function will just evaluate to 2, which is right there. But as soon as we hit t is equal to pi, pi is a c in this example, as soon as we hit that, the unit step function becomes 1.... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So our function will just evaluate to 2, which is right there. But as soon as we hit t is equal to pi, pi is a c in this example, as soon as we hit that, the unit step function becomes 1. We multiply that by 2, and we have 2 minus 2, and then we end up here with 0. Now, that might be nice and everything, but let's say ... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | Now, that might be nice and everything, but let's say you wanted for it to go back up. Let's say that instead of it going like this, let me kind of erase that by overdrawing the x-axis again. We want the function to jump up again. Let's say it's some value, let's say it's at 2 pi. We want the function to jump up again.... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | Let's say it's some value, let's say it's at 2 pi. We want the function to jump up again. How could we construct this? We could make it jump to anything. Let's say we want it to jump back to 2. Well, we could just add another unit step function here, something that would have been 0 all along, all the way up until this... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | We could make it jump to anything. Let's say we want it to jump back to 2. Well, we could just add another unit step function here, something that would have been 0 all along, all the way up until this point. But then at 2 pi, it jumps. So it would be, in this case, our c would be 2 pi. So unit step function. We want i... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | But then at 2 pi, it jumps. So it would be, in this case, our c would be 2 pi. So unit step function. We want it to jump to 2. This would just jump to 1 by itself, so let's multiply it by 2. And now we have this function. So you could imagine, you could make an arbitrarily complicated function of things jumping up and ... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | We want it to jump to 2. This would just jump to 1 by itself, so let's multiply it by 2. And now we have this function. So you could imagine, you could make an arbitrarily complicated function of things jumping up and down to different levels based on different, essentially linear combinations of these unit step functi... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So you could imagine, you could make an arbitrarily complicated function of things jumping up and down to different levels based on different, essentially linear combinations of these unit step functions. Now, what if I wanted to do something a little bit fancier? What if I wanted to do something that, let's say I have... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | Let me draw some function. I should draw straighter than that. I should have some standards. So let's say that just my regular f of t, this is x. Actually, why am I doing x? This should be the t-axis. We're doing the time domain. |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So let's say that just my regular f of t, this is x. Actually, why am I doing x? This should be the t-axis. We're doing the time domain. It could have been x. And then this is, we could call this f of t. So let me draw some arbitrary f of t. Let's say my function looks something crazy like that. Now, what if I'm modeli... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | We're doing the time domain. It could have been x. And then this is, we could call this f of t. So let me draw some arbitrary f of t. Let's say my function looks something crazy like that. Now, what if I'm modeling a physical system? This is my f of t. What if I'm modeling a physical system that doesn't do this, that a... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | Now, what if I'm modeling a physical system? This is my f of t. What if I'm modeling a physical system that doesn't do this, that actually, at some point, well, actually, let's say it stays at 0 until some value. Let's say it goes to 0 until, I'll call that c again. And then at c, f of t kind of starts up. So right at ... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | And then at c, f of t kind of starts up. So right at c, f of t should start up. So it just kind of goes like this. So essentially, what we have here is a combination of it's 0 all the way, and then we have a shifted f of t. So at c, we have a shifted f of t. So it shifts that way. So how can we construct this yellow fu... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So essentially, what we have here is a combination of it's 0 all the way, and then we have a shifted f of t. So at c, we have a shifted f of t. So it shifts that way. So how can we construct this yellow function, where it's essentially a shifted version of this green function, but it's 0 below c? This green function mi... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | It might have gone something like this. It might have continued and done something crazy. But what we did is we shifted it from here to there, and then we zeroed out everything before c. So how could we do that? Well, just shifting this function you've learned in your Algebra 2 or your Precalculus classes, to shift a f... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | Well, just shifting this function you've learned in your Algebra 2 or your Precalculus classes, to shift a function by c to the right, you just replace your t with a t minus c. So this function right here is f of t minus c. And to make sure I get it right, what I always do is I imagine, OK, what's going to happen when ... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So when t is equal to c, this value, the value of the function should be equivalent to the value of the original green function at 0. So it's equivalent to that value, which makes sense. If we go up one more above c, so let's say this is one more above c, so we get to this point. When you, if t is c plus 1, then when y... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | When you, if t is c plus 1, then when you put c plus 1 minus c, you just have f of 1. And f of 1 is really just this point right here. And so it'll be that f of 1. So it makes sense. So as we move one forward here, we're essentially at the same function value as we were there. So the shift works. But I said that we had... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So it makes sense. So as we move one forward here, we're essentially at the same function value as we were there. So the shift works. But I said that we had to also, if I just shifted this function, you would have all this other stuff. Because you would have had all this other stuff when the function was back here, sti... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | But I said that we had to also, if I just shifted this function, you would have all this other stuff. Because you would have had all this other stuff when the function was back here, still going on. The function, I'll draw it lightly, would still continue. But I said I wanted to zero out this function before we reach c... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | But I said I wanted to zero out this function before we reach c. So how can I zero out that function? Well, I think it's pretty obvious to you. I started this video talking about the unit step function. So what if I multiply the unit step function times this thing, what's going to happen? So what if I, my new function,... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So what if I multiply the unit step function times this thing, what's going to happen? So what if I, my new function, I call it the unit step function up till c of t, times f of t minus c. So what's going to happen? Until we get to c, the unit step function is 0 when it's less than c. So you're going to have 0 times, I... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So this function is going to be 0. Once you hit c, the unit step function becomes 1. So once you pass c, this thing becomes a 1. And you're just left with 1 times your function. So then your function can behave as it would like to behave. And you actually shifted it. This t minus c is what actually shifted this green f... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | And you're just left with 1 times your function. So then your function can behave as it would like to behave. And you actually shifted it. This t minus c is what actually shifted this green function over to the right. And this is actually going to be a very useful way to look at, or a very usefully constructed function... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | This t minus c is what actually shifted this green function over to the right. And this is actually going to be a very useful way to look at, or a very usefully constructed function. And in a second, we're going to figure out the Laplace transform of this. And you're going to appreciate, I think, why this is a useful f... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | And you're going to appreciate, I think, why this is a useful function to look at. But now you understand at least what it is and why it essentially shifts a function and zeros out everything before that point. Well, I told you that this is a useful function. So we should add its Laplace transform to our library of Lap... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So we should add its Laplace transform to our library of Laplace transforms. So let's do that. So let's take the Laplace transform of this, of the unit step function up to c. I'm doing it in fairly general terms. In the next video, we'll do a bunch of examples where we can apply this. But we should at least prove to ou... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | In the next video, we'll do a bunch of examples where we can apply this. But we should at least prove to ourselves what the Laplace transform of this thing is. Well, the Laplace transform of anything, or our definition of it so far, is the integral from 0 to infinity of e to the minus st times our function. So our func... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So our function in this case is the unit step function, u sub c of t times f of t minus c dt. And it seems very general. It seems very hard to evaluate this integral at first. But maybe we can make some form of a substitution to get it into a term that we can appreciate. So let's make a substitution here. Let me pick a... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | But maybe we can make some form of a substitution to get it into a term that we can appreciate. So let's make a substitution here. Let me pick a nice variable to work with. I don't know. We're not using an x anywhere. We might as well use an x. That's the most fun variable to work with. |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | I don't know. We're not using an x anywhere. We might as well use an x. That's the most fun variable to work with. x. Sometimes you'll see in a lot of math classes they introduce these crazy Latin alphabets. And that by itself makes it hard to understand. |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | That's the most fun variable to work with. x. Sometimes you'll see in a lot of math classes they introduce these crazy Latin alphabets. And that by itself makes it hard to understand. So I like to stay away from those crazy Latin alphabets. So we'll just use a regular x. Let's make a substitution. |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | And that by itself makes it hard to understand. So I like to stay away from those crazy Latin alphabets. So we'll just use a regular x. Let's make a substitution. Let's say that x is equal to t minus c. Or if we added t to both sides, we could say that t is equal to x plus c. Let's see what happens to our substitution.... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | Let's make a substitution. Let's say that x is equal to t minus c. Or if we added t to both sides, we could say that t is equal to x plus c. Let's see what happens to our substitution. And also if we took the derivative of both sides of this, or I guess the differential, you would get dx is equal to dt. Or I mean if yo... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | Or I mean if you took dx with respect to dt, you would get that to be equal to 1. c is just a constant. And if you multiply both sides by dt, you get dx is equal to dt, and that's a nice substitution. So what is our integral going to become with this substitution? So our integral, this was t equals 0 to t is equal to i... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So our integral, this was t equals 0 to t is equal to infinity. When t is equal to 0, what is x going to be equal to? Well, x is going to be equal to minus c. Actually, before I go there, let me actually take a step back, because we could go in this direction. We could actually simplify it more before we do that. Let's... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | We could actually simplify it more before we do that. Let's go back to our original integral before we even made our substitution. If we're taking the integral from 0 to infinity of this thing, we already said, what does this integral look like? Or what does this function look like? It's 0. We have this unit step funct... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | Or what does this function look like? It's 0. We have this unit step function sitting right here. We have the unit step function sitting right there. So this whole expression is going to be 0 until we get to c. This whole thing, by definition, this unit step function is 0 until we get to c. So everything's going to be ... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | We have the unit step function sitting right there. So this whole expression is going to be 0 until we get to c. This whole thing, by definition, this unit step function is 0 until we get to c. So everything's going to be zeroed out until we get to c. So we could essentially say, we don't have to take the integral from... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | I'll just use that old integral sign. We could just take the integral from t is equal to c to t is equal to infinity of e to the minus st, the unit step function, uc of t times f of t minus c dt. In fact, at this point, this unit step function has no use anymore because before t is equal to c, it's 0. And now that we'r... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | And now that we're only worried about values above c, it's equal to 1. So it equals 1 in this context. I want to make that very clear to you. What did I do just here? I changed our bottom boundary from 0 to c. And I think you might realize why I did it when I was working with the substitution. Because this will simplif... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | What did I do just here? I changed our bottom boundary from 0 to c. And I think you might realize why I did it when I was working with the substitution. Because this will simplify things if we do this ahead of time. So if we have this unit step function, this thing is going to zero out this entire integral before we ge... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So if we have this unit step function, this thing is going to zero out this entire integral before we get to c. Remember, this definite integral is really just the area under this curve, right? Of this whole function, of the unit step function times all of this stuff. All of this stuff, when we multiply it, it's going ... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So if we want to essentially find the area under this curve, we can ignore all the stuff that happens before c. So instead of going from t equals 0 to infinity, we can go from t is equal to c to infinity. Because there was no area before t was equal to c. So that's all I did here. And then the other thing I said is tha... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So we can just kind of ignore it, right? It's just going to be 1 this entire time. So our integral simplifies to the definite integral from t is equal to c to t is equal to infinity of e to the minus st times f of t minus c dt. And this will simplify it a good bit. I was going down the other road when I did the substit... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | And this will simplify it a good bit. I was going down the other road when I did the substitution first, which would have worked. But I think the argument as to why I could have changed the boundaries would have been a harder argument to make. So now that we had this, let's go back and make that substitution that x is ... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So now that we had this, let's go back and make that substitution that x is equal to t minus c. So our integral becomes, I'll do it in green, when t is equal to c, what is x? Then x is 0, right? c minus c is 0. When t is equal to infinity, what is x? Well, x is infinity minus any constant. It's still going to be infini... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | When t is equal to infinity, what is x? Well, x is infinity minus any constant. It's still going to be infinity. Or the limit as t approaches infinity, x is still going to be infinity here. And it's the integral of e to the minus s. But now instead of a t, we have the substitution. If we said x is equal to t minus c, t... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | Or the limit as t approaches infinity, x is still going to be infinity here. And it's the integral of e to the minus s. But now instead of a t, we have the substitution. If we said x is equal to t minus c, then we can just add c to both sides. You get t is equal to x plus c. So you get x plus c there. And then times a ... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | You get t is equal to x plus c. So you get x plus c there. And then times a function, f of t minus c. But we said t minus c is the same thing as x. And dt is the same thing as dx. Showed you that right there. So we can write this as dx. Now this is starting to look a little bit interesting. So what is this equal to? |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | Showed you that right there. So we can write this as dx. Now this is starting to look a little bit interesting. So what is this equal to? This is equal to the integral from 0 to infinity. Let me expand this out of e to the minus sx minus sc times f of x dx. Now what is this equal to? |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So what is this equal to? This is equal to the integral from 0 to infinity. Let me expand this out of e to the minus sx minus sc times f of x dx. Now what is this equal to? Well, we could factor out an e to the minus sc and bring it outside of the integral. Because this has nothing to do with what we're taking the inte... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | Now what is this equal to? Well, we could factor out an e to the minus sc and bring it outside of the integral. Because this has nothing to do with what we're taking the integral with respect to. So let's do that. Let me take this guy out. And just to not confuse you, let me rewrite the whole thing, 0 to infinity. I co... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So let's do that. Let me take this guy out. And just to not confuse you, let me rewrite the whole thing, 0 to infinity. I could rewrite this e term as e to the minus sx times e to the minus sc, right? Common base. So if I were to multiply these two, I could just add the exponents, which you would get that up there, tim... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | I could rewrite this e term as e to the minus sx times e to the minus sc, right? Common base. So if I were to multiply these two, I could just add the exponents, which you would get that up there, times f of x dx. This is a constant term with respect to x. So we can just factor it out. We can just factor this thing out... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | This is a constant term with respect to x. So we can just factor it out. We can just factor this thing out right there. So then you get e to the minus sc times the integral from 0 to infinity of e to the minus sx times f of x dx. Now what were we doing here the whole time? We were taking the Laplace transform of the un... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | So then you get e to the minus sc times the integral from 0 to infinity of e to the minus sx times f of x dx. Now what were we doing here the whole time? We were taking the Laplace transform of the unit step function that goes up to c, and then it's 0 up to c, and it's 1 after that, of t times some shifted function, f ... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | And we made a substitution. We simplified it a little bit. e to the minus sc times the integral from 0 to infinity of e to the minus sx f of x dx. Now something about the tablet doesn't work properly right around this period. But this should look interesting to you. What is this? This is the Laplace transform of f of x... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | Now something about the tablet doesn't work properly right around this period. But this should look interesting to you. What is this? This is the Laplace transform of f of x. Let me write that down. What's the Laplace transform of, well, I could write it as f of t or f of x. The Laplace transform of f of t is equal to ... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | This is the Laplace transform of f of x. Let me write that down. What's the Laplace transform of, well, I could write it as f of t or f of x. The Laplace transform of f of t is equal to the integral from 0 to infinity of e to the minus st times f of t dt. I mean, this and this are the exact same thing. We're just using... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | The Laplace transform of f of t is equal to the integral from 0 to infinity of e to the minus st times f of t dt. I mean, this and this are the exact same thing. We're just using a t here. We're using an x here. No difference. They're just letters. But they're, oh, let me, this is f of t. e to the minus st times f of t... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | We're using an x here. No difference. They're just letters. But they're, oh, let me, this is f of t. e to the minus st times f of t dt. I could have also rewritten it as the Laplace transform of f of t. I could write this as the integral from 0 to infinity of e to the minus sy times f of y dy. I could do it by anything... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | But they're, oh, let me, this is f of t. e to the minus st times f of t dt. I could have also rewritten it as the Laplace transform of f of t. I could write this as the integral from 0 to infinity of e to the minus sy times f of y dy. I could do it by anything because this is a definite integral. The y's are going to d... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | The y's are going to disappear. And we've seen that. All you're left with is a function of s. This ends up being some capital, well, we could write some capital function of s. So this is interesting. This is a Laplace transform of f of t times some scaling factor, and that's what we set out to show. So we can now show ... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | This is a Laplace transform of f of t times some scaling factor, and that's what we set out to show. So we can now show that the Laplace transform of the unit step function times some function t minus c is equal to this function right here, e to the minus sc, where this c is the same as this c right here, times the Lap... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | Let's see if we backfilled it somehow. What does this mean? What can we do with this? Well, let's say we had the Laplace transform. Let's say we wanted to figure out the Laplace transform of the unit step function that starts off at pi of t. And let's say we're taking, I don't know, something that we know well, sine of... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | Well, let's say we had the Laplace transform. Let's say we wanted to figure out the Laplace transform of the unit step function that starts off at pi of t. And let's say we're taking, I don't know, something that we know well, sine of t minus pi. So we shifted it, right? This thing is really malfunctioning at this poin... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | This thing is really malfunctioning at this point right here. Let me pause it. I just paused, sorry if that was a little disconcerting. I just paused the video because it was having trouble recording at some point on my little board. So let me rewrite the result that we proved just now. We showed that the Laplace trans... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | I just paused the video because it was having trouble recording at some point on my little board. So let me rewrite the result that we proved just now. We showed that the Laplace transform of the unit step function, t, and it goes to 1 at some value c, times some function that's shifted by c to the right. It's equal to... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | It's equal to e to the minus cs times the Laplace transform of just the unshifted function. That was our result. That was a big takeaway from this video. And if this seems like some Byzantine, hard to understand result, we can apply it. So let's say the Laplace transform, and this is what I was doing right before the a... |
Laplace transform of the unit step function Laplace transform Khan Academy.mp3 | And if this seems like some Byzantine, hard to understand result, we can apply it. So let's say the Laplace transform, and this is what I was doing right before the actual pen tablet started malfunctioning. If we want to take the Laplace transform of the unit step function that goes to 1 at pi, t, times the sine functi... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | Let's do some more examples with exact differential equations. And I'm getting these problems from page 80 of my old college differential equations book. This is the fifth edition of Elementary Differential Equations by William Boyce and Richard DePrima. I want to make sure they get credit that I'm not making up these ... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | I want to make sure they get credit that I'm not making up these problems. I'm getting it from their book. Anyway, so I'm just going to give a bunch of equations. We have to figure out if they're exact. And if they are exact, we'll use what we know about exact differential equations to figure out their solutions. So th... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | We have to figure out if they're exact. And if they are exact, we'll use what we know about exact differential equations to figure out their solutions. So the first one they have is 2x plus 3 plus 2y minus 2 times y prime is equal to 0. So this is our m of x and y, although this is only a function of x. And this is our... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | So this is our m of x and y, although this is only a function of x. And this is our n. You could say that's m or that's n. You could also say that if this is exact, well, first let's just say it's exact before we start talking about xi. So what's the partial of this with respect to y? The partial of m with respect to y... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | The partial of m with respect to y. Well, there's no y here, so it's 0. The rate of change that this changes with respect to y is 0. And what's the rate of change that this changes with respect to x? The partial of n with respect to x is equal to, well, there's no x here, right? So these are just constants from an x po... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | And what's the rate of change that this changes with respect to x? The partial of n with respect to x is equal to, well, there's no x here, right? So these are just constants from an x point of view, so this is all going to be 0. But we do see that they're both 0. So m sub y, or the partial with respect to y, is equal ... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | But we do see that they're both 0. So m sub y, or the partial with respect to y, is equal to the partial with respect to x. So this is exact. And actually, we don't have to use exact equations here. We'll do it just so that we get used to it. But if you looked here, you actually could have figured out that this is actu... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | And actually, we don't have to use exact equations here. We'll do it just so that we get used to it. But if you looked here, you actually could have figured out that this is actually a separable equation. But anyway, this is exact. So knowing that it's exact, it tells us there's some function xi, where xi is a function... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | But anyway, this is exact. So knowing that it's exact, it tells us there's some function xi, where xi is a function of x and y, where xi sub x is equal to this function, is equal to 2x plus 3. And xi, I shouldn't say sub x, I should say the partial of xi with respect to x. And the partial of xi with respect to y is equ... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | And the partial of xi with respect to y is equal to this, 2y minus 2. And if we can find our xi, we know that this is just the derivative of xi, right? Because we know that the derivative with respect to x of xi is equal to the partial of xi with respect to x plus the partial of xi with respect to y times y prime. So t... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | So this is just the same form as that. So if we can figure out y, then we could rewrite this equation as dx, the derivative of xi with respect to x, is equal to 0, right? This, let me switch colors just because it gets monotonous. This right here, if we can find a xi where the partial with respect to x is this, the par... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | This right here, if we can find a xi where the partial with respect to x is this, the partial with respect to y is this, then this can be rewritten as this, right? And how do we know that? Because the derivative of xi with respect to x, using the partial derivative chain rules, is this. And this is the same. This parti... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | And this is the same. This partial with respect to x, that's this. Partial with respect to y is this times y prime. So this is a whole point of exact equations. But anyway, so let's figure out what our xi is. Actually, before we figure out, if the derivative of xi with respect to x is 0, then if you integrate both side... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | So this is a whole point of exact equations. But anyway, so let's figure out what our xi is. Actually, before we figure out, if the derivative of xi with respect to x is 0, then if you integrate both sides, the solution of this equation is xi is equal to c. So using this information, if we can solve for xi, then we kno... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | So let's integrate both sides of this equation with respect to x. And then we get xi is equal to x squared plus 3x plus some function of y. Let's call it h of y. And remember, normally when you take an antiderivative, you have just a plus c here, right? But you can kind of say we took an antipartial derivative. So when... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | And remember, normally when you take an antiderivative, you have just a plus c here, right? But you can kind of say we took an antipartial derivative. So when you took a partial derivative with respect to x, not only do you lose constants, that's why we have a plus c normally, but you also lose anything that's a functi... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | So for example, take the partial derivative of this with respect to x. You're going to get this, right? Because the partial derivative of a function purely of y with respect to x is going to be 0, so it will disappear. So anyway, we take the antiderivative of this, we get this. Now we use this information. We take the ... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | So anyway, we take the antiderivative of this, we get this. Now we use this information. We take the partial of this expression, and we say, well, the partial of this expression with respect to y has to equal this, and then we can solve for h of y, and we'll be done. So let's do that. So the partial of xi with respect ... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | So let's do that. So the partial of xi with respect to y is equal to, well, that's going to be 0, 0, 0. It's going to be, right? This part is a function of x. You take the partial with respect to y, it's 0, because these are constants from a y point of view. So you're left with h prime of y. So we know that h prime of ... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | This part is a function of x. You take the partial with respect to y, it's 0, because these are constants from a y point of view. So you're left with h prime of y. So we know that h prime of y, which is the partial of xi with respect to y, is equal to this. So h prime of y is equal to 2y minus 2. And then if we wanted ... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | So we know that h prime of y, which is the partial of xi with respect to y, is equal to this. So h prime of y is equal to 2y minus 2. And then if we wanted to figure out what h of y is, we get h of y, just integrate both sides with respect to y, is equal to y squared minus 2y. Now you could have a plus c there, but if ... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | Now you could have a plus c there, but if you watch the previous example, you'll see that that c kind of merges with the other c, so you don't have to worry about it right now. So what is our xi function, as we know it now, not worrying about the plus c? It is xi of x and y is equal to x squared plus 3x plus h of y, wh... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | And we know a solution of our original differential equation is xi is equal to c. So the solution of our differential equation is this is equal to c. It's x squared plus 3x plus y squared minus 2y is equal to c. And if you had some initial conditions, you could test it. And I encourage you to test this out on this orig... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | The more examples you see, the better. So let's see, this one says 2x plus 4y plus 2x minus 2y, y prime is equal to 0. So what's the partial of this with respect to y? So m, the partial of m with respect to y, this is 0, so it's equal to 4. What's the partial of this with respect to x, just this part right here? The pa... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | So m, the partial of m with respect to y, this is 0, so it's equal to 4. What's the partial of this with respect to x, just this part right here? The partial of n with respect to x is 2, this is 0. So the partial of this with respect to y is different than the partial of n with respect to x. So this is not exact. So we... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | So the partial of this with respect to y is different than the partial of n with respect to x. So this is not exact. So we can't solve this using our exact methodology. So that was a fairly straightforward problem. Let's do another one. Let's see, I'm running out of time, so I want to do one that's not too complicated.... |
Exact equations example 2 First order differential equations Khan Academy.mp3 | So that was a fairly straightforward problem. Let's do another one. Let's see, I'm running out of time, so I want to do one that's not too complicated. Let's see, we have 3x squared minus 2xy. Actually, let me do this in the next problem. I don't want to rush these things. I will continue this in the next video. |
Laplace as linear operator and Laplace of derivatives Laplace transform Khan Academy.mp3 | And the first is to show that it is a linear operator. What does that mean? Well, let's say I wanted to take the Laplace transform of the sum of the, we call it the weighted sum of two functions. So say some constant, c1 times my first function, f of t, plus some constant, c2, times my second function, g of t. Well, by... |
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