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Old separable differential equations introduction Khan Academy.mp3 | So let's multiply both sides by 2 times y minus 1. And you get 2 times y minus 1 times dy dx is equal to 3x squared plus 4x plus 2. Multiply both sides times dx. This is really just an exercise in algebra. You get... and I can multiply this one out too. You get 2y minus 2, that's just this, dy. I multiply both sides ti... |
Old separable differential equations introduction Khan Academy.mp3 | This is really just an exercise in algebra. You get... and I can multiply this one out too. You get 2y minus 2, that's just this, dy. I multiply both sides times dx, so that equals 3x squared plus 4x plus 2 dx. I have separated the equations. I've separated the independent from the dependent variable and their relative... |
Old separable differential equations introduction Khan Academy.mp3 | I multiply both sides times dx, so that equals 3x squared plus 4x plus 2 dx. I have separated the equations. I've separated the independent from the dependent variable and their relative differentials. And so now I can integrate. And I can integrate in magenta. I can integrate. What's the antiderivative of this express... |
Old separable differential equations introduction Khan Academy.mp3 | And so now I can integrate. And I can integrate in magenta. I can integrate. What's the antiderivative of this expression with respect to y? Well, let's just see. It's y squared minus 2y. I won't write the plus C. I'll just do it on the right-hand side. |
Old separable differential equations introduction Khan Academy.mp3 | What's the antiderivative of this expression with respect to y? Well, let's just see. It's y squared minus 2y. I won't write the plus C. I'll just do it on the right-hand side. That is equal to 3x squared. Well, the antiderivative is x to the third plus antiderivative is 2x squared plus 2x plus C. And that C kind of ta... |
Old separable differential equations introduction Khan Academy.mp3 | I won't write the plus C. I'll just do it on the right-hand side. That is equal to 3x squared. Well, the antiderivative is x to the third plus antiderivative is 2x squared plus 2x plus C. And that C kind of takes care of the constant for both sides of the equation. And hopefully you understand y from the last example. ... |
Old separable differential equations introduction Khan Academy.mp3 | And hopefully you understand y from the last example. But we can solve for C using the initial condition y of 0 is equal to negative 1. So let's see. When x is 0, y is negative 1. So let's put y as negative 1. So we get negative 1 squared minus 2 times negative 1. That's the value of y. |
Old separable differential equations introduction Khan Academy.mp3 | When x is 0, y is negative 1. So let's put y as negative 1. So we get negative 1 squared minus 2 times negative 1. That's the value of y. It's equal to when x is equal to 0. So when x is equal to 0, that's 0 to the third plus 2 times 0 squared plus 2 times 0 plus C. So this is fairly straightforward. All of these, this... |
Old separable differential equations introduction Khan Academy.mp3 | That's the value of y. It's equal to when x is equal to 0. So when x is equal to 0, that's 0 to the third plus 2 times 0 squared plus 2 times 0 plus C. So this is fairly straightforward. All of these, this is all 0. This is, let's see, negative 1 squared. That's 1 minus 2 times minus 1. That's plus 2. |
Old separable differential equations introduction Khan Academy.mp3 | All of these, this is all 0. This is, let's see, negative 1 squared. That's 1 minus 2 times minus 1. That's plus 2. It's equal to C. And we get C is equal to 3. So the implicit exact solution, the solution of our differential equation, remember now it's not a class because they gave us an initial condition, is y square... |
Old separable differential equations introduction Khan Academy.mp3 | That's plus 2. It's equal to C. And we get C is equal to 3. So the implicit exact solution, the solution of our differential equation, remember now it's not a class because they gave us an initial condition, is y squared minus 2y is equal to x to the third plus 2x squared plus 2x plus 3. We figured out that's what C wa... |
Old separable differential equations introduction Khan Academy.mp3 | We figured out that's what C was. And actually if you want, you could write this in an explicit form by completing the square. This is just algebra at this point. You're done. This is an implicit form. If you wanted to make it explicit, you could add 1 to both sides. I'm just completing the square here. |
Old separable differential equations introduction Khan Academy.mp3 | You're done. This is an implicit form. If you wanted to make it explicit, you could add 1 to both sides. I'm just completing the square here. So y squared minus 2y plus 1. If I add 1 to that side, I'd have to add 1 to this side. So it becomes x to the third plus 2x squared plus 2x plus 4. |
Old separable differential equations introduction Khan Academy.mp3 | I'm just completing the square here. So y squared minus 2y plus 1. If I add 1 to that side, I'd have to add 1 to this side. So it becomes x to the third plus 2x squared plus 2x plus 4. I just added 1 to both sides of this equation. Why did I do that? Because I wanted this side to be a perfect square in terms of y. |
Old separable differential equations introduction Khan Academy.mp3 | So it becomes x to the third plus 2x squared plus 2x plus 4. I just added 1 to both sides of this equation. Why did I do that? Because I wanted this side to be a perfect square in terms of y. Then I could rewrite this side as y minus 1 squared is equal to x to the third plus 2x squared plus 2x plus 4. Then I could say ... |
Old separable differential equations introduction Khan Academy.mp3 | Because I wanted this side to be a perfect square in terms of y. Then I could rewrite this side as y minus 1 squared is equal to x to the third plus 2x squared plus 2x plus 4. Then I could say y minus 1 is equal to the plus or minus square root of x to the third plus 2x squared plus 2x plus 4. I could add 1 to both sid... |
Old separable differential equations introduction Khan Academy.mp3 | I could add 1 to both sides and then I could get y is equal to 1 plus or minus the square root of x to the third plus 2x squared plus 2x plus 4. It has plus or minus here. If we had to pick one of the two, we'd go back to the initial condition. Our initial condition told us that y of 0 is equal to negative 1. If we put... |
Old separable differential equations introduction Khan Academy.mp3 | Our initial condition told us that y of 0 is equal to negative 1. If we put 0 here for x, we get y is equal to 1 plus or minus 0 plus 4. So 1 plus or minus 4. If y is going to be equal to negative 1, so we get y is equal to 1 plus or minus 2. If this is going to be equal to negative 1, then this has to be 1 minus 2. So... |
Old separable differential equations introduction Khan Academy.mp3 | If y is going to be equal to negative 1, so we get y is equal to 1 plus or minus 2. If this is going to be equal to negative 1, then this has to be 1 minus 2. So the explicit form that satisfies our initial condition, and we're getting a little geeky here, you can get rid of the plus. It's 1 minus this whole thing. Tha... |
Old separable differential equations introduction Khan Academy.mp3 | It's 1 minus this whole thing. That's what satisfies our initial condition. You could figure out where it's satisfied because in order for this, over what domain is it satisfied? That's satisfied when this term is positive. This becomes negative and you get it undefined in reals and all of that. Anyway, I've run out of... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | And if I wanted to take the derivative of this with respect to x, this is equal to the partial of xi with respect to x plus the partial of xi with respect to y times dy dx. And in the last video, I didn't prove it to you, but I hopefully gave you a little bit of intuition that you can believe me. But maybe one day I'll... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | But you can find proofs on the web if you're interested for the chain rule with partial derivatives. So let's put that aside and let's explore another property of partial derivatives. And then we're ready to get the intuition behind exact equations. Because what you're going to find, it's fairly straightforward to solv... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | Because what you're going to find, it's fairly straightforward to solve exact equations. But the intuition is a little bit more, well, I don't want to say it's difficult. Because if you have the intuition, you have it. So what if I had, say, the same function xi, and I were to take the partial derivative of xi with res... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | So what if I had, say, the same function xi, and I were to take the partial derivative of xi with respect to x first. I'll just write xi. I don't have to write x and y every time. And then I were to take the partial derivative with respect to y. So just as a notation, this you could write as, you could kind of view it ... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | And then I were to take the partial derivative with respect to y. So just as a notation, this you could write as, you could kind of view it as you're multiplying the operators. So it could be written like this. The partial del squared times xi, or del squared xi, over del y, del, or curly dx. And that can also be writt... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | The partial del squared times xi, or del squared xi, over del y, del, or curly dx. And that can also be written as, and this is my preferred notation, because it doesn't have all this extra junk everywhere. You could just say, well, we took the partial with respect to x first. So this just means the partial of xi with ... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | So this just means the partial of xi with respect to x. And then we took the partial with respect to y. So that's one situation to consider. What happens when we take the partial with respect to x and then y? So with respect to x, you hold y constant to get just the partial with respect to x, ignore the y there. And th... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | What happens when we take the partial with respect to x and then y? So with respect to x, you hold y constant to get just the partial with respect to x, ignore the y there. And then you hold the x constant, and you take the partial with respect to y. So what's the difference between that and if we were to switch the or... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | So what's the difference between that and if we were to switch the order? So what happens if we were to, I'll do it in a different color. If we had xi, and we were to take the partial with respect to y first, and then we were to take the partial with respect to x. So just the notation, just you're comfortable with it. ... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | So just the notation, just you're comfortable with it. That would be, so partial x, partial y. And this is the operator. And it might be a little confusing that here between these two notations, even though they're the same thing, the order is mixed. That's just because it's just a different way of thinking about it. T... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | And it might be a little confusing that here between these two notations, even though they're the same thing, the order is mixed. That's just because it's just a different way of thinking about it. This says, OK, partial first with respect to x, then y. This views it more as the operator. So we took the partial of x fi... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | This views it more as the operator. So we took the partial of x first, and then we took y. Like you're multiplying the operators. But anyway, so this can also be written as the partial of y, and then we took the partial of that with respect to x. Now I'm going to tell you right now that if each of the first partials ar... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | But anyway, so this can also be written as the partial of y, and then we took the partial of that with respect to x. Now I'm going to tell you right now that if each of the first partials are continuous, and most of the functions we've dealt with in a normal domain, as long as there aren't any discontinuities or holes ... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | If both of these functions are continuous, if both of the first partials are continuous, then these two are going to be equal to each other. So xi of xy is going to be equal to xi of yx. Now, we can use this knowledge, which is the chain rule using partial derivatives, and this knowledge to now solve a certain class of... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | And what does an exact equation look like? An exact equation looks like this. It's always the color picking is the hard part. So let's say I have, this is my differential equation. I have some function of x and y. So I don't know, it could be x squared times cosine of y or something, I don't know. It could be any funct... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | So let's say I have, this is my differential equation. I have some function of x and y. So I don't know, it could be x squared times cosine of y or something, I don't know. It could be any function of x and y. Plus some function of x and y, we'll call that n, times dy dx is equal to 0. This is, well, I don't know it's ... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | It could be any function of x and y. Plus some function of x and y, we'll call that n, times dy dx is equal to 0. This is, well, I don't know it's an exact equation yet, but if you saw something of this form, your first impulse should be, oh, well, actually, your very first impulse is, is this separable? And you should... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | And you should try to play around with the algebra a little bit and see if it's separable, because that's always the most straightforward way. If it's not separable, but you can still put it in this form, you say, hey, is it an exact equation? And what's an exact equation? Well, look immediately. This pattern right her... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | Well, look immediately. This pattern right here looks an awful lot like this pattern. What if m was the partial of xi with respect to x? What if xi with respect to x is equal to m? What if this was xi with respect to x, and what if this was xi with respect to y? So xi with respect to y is equal to n. What if? I'm just ... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | What if xi with respect to x is equal to m? What if this was xi with respect to x, and what if this was xi with respect to y? So xi with respect to y is equal to n. What if? I'm just saying, we don't know for sure, right? If you just see this someplace randomly, you won't know for sure that this is the partial with res... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | I'm just saying, we don't know for sure, right? If you just see this someplace randomly, you won't know for sure that this is the partial with respect to x of some function, and this is the partial with respect to y of some function. But we're just saying, what if? If this were true, then we could rewrite this as the p... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | If this were true, then we could rewrite this as the partial of xi with respect to x plus the partial of xi with respect to y times dy dx equal to 0. And this right here, the left side right there, that's the same thing as this. This is just the derivative of xi with respect to x using the partial derivative chain rule... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | So you could rewrite it. This is just the derivative of xi with respect to x. And xi is a function of xy is equal to 0. So if you see a differential equation, it has this form, and you're saying, boy, I can't separate it, but maybe it's an exact equation. And frankly, if that was what was recently covered before the cu... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | So if you see a differential equation, it has this form, and you're saying, boy, I can't separate it, but maybe it's an exact equation. And frankly, if that was what was recently covered before the current exam, it probably is an exact equation. But if you see this form, you say, boy, maybe it's an exact equation. If i... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | If it is an exact equation, and I'll show you how to test it in a second using this information, then this can be written as the derivative of some function xi, where this is the partial of xi with respect to x. This is the partial of xi with respect to y. And then if you could write it like this, and you take the deri... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | So there are two things that we should be caring about that you might be saying, OK, Sal, you've walked through xi's and partials and all this. One, how do I know that it's an exact equation? And then if it is an exact equation, which tells us that there is some xi, then how do I solve for the xi? So the way to figure ... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | So the way to figure out is it an exact equation is to use this information right here. We know that if xi and its derivatives are continuous over some domain, that when you take the partial with respect to x and then y, that's the same thing as doing it in the other order. So we said, this is the partial with respect ... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | This is the partial with respect to x, and this is the partial with respect to y. So if this is an exact equation, if we were to take the partial of this with respect to y, if we were to say, if we were to take the partial of m with respect to y, so the partial of xi with respect to x is equal to m. If we were to take ... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | So then this will also be equal. So that is actually the test to test if this is an exact equation. So let me rewrite all of that again and summarize it a little bit. So if you see something of the form m of xy plus n of xy times dy dx is equal to 0. And then you take the partial derivative of m with respect to y, and ... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | So if you see something of the form m of xy plus n of xy times dy dx is equal to 0. And then you take the partial derivative of m with respect to y, and then you take the partial derivative of n with respect to x, and they are equal to each other, then, and it's actually if and only if, so it goes both ways, this is an... |
Exact equations intuition 2 (proofy) First order differential equations Khan Academy.mp3 | And if it's an exact equation, that tells us that there exists a xi such that the derivative of xi of xy is equal to 0, or xi of xy is equal to c is a solution of this equation. And the partial derivative of xi with respect to x is equal to m, and the partial derivative of xi with respect to y is equal to n. And I'll s... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | OK, I filled your brain with a bunch of partial derivatives and xi's with respect to x's and y's. I think now it's time to actually do it with a real differential equation and make things a little bit more concrete. So let's say I have the differential equation y cosine of x plus 2xe to the y plus sine of x plus x squa... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | Well, you could probably already, your brain is already, hopefully, in exact differential equations mode. But if you were to just see this pattern in general, where you see a function of x and y here, this is just some function of x and y. And then you have another function of x and y times y prime or times dy d of x. ... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | Your brain should immediately say if this isn't separable, and I'm not going to try to make it separable, just because that'll take a lot of time. But if it's not separable, your brain said, oh, maybe this is an exact equation. And you say, let me test whether this is an exact equation. So if this is an exact equation,... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So if this is an exact equation, this is our function m, which is a function of x and y. And this is our function n, which is a function of x and y. Now the test is to see if the partial of this with respect to y is equal to the partial of this with respect to x. So let's see. The partial of m with respect to y is equa... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So let's see. The partial of m with respect to y is equal to, let's see, y is, so this cosine of x is just a constant, so it's just cosine of x plus. Now, what's the derivative? Well, 2x is just a constant. What's the derivative of e to the y with respect to y? Well, it's just e to the y, right? So we have the constant... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | Well, 2x is just a constant. What's the derivative of e to the y with respect to y? Well, it's just e to the y, right? So we have the constant on the outside, 2x times the derivative with respect to y, so it's 2x e to the y. Fair enough. Now what is the partial derivative of this with respect to x? So n sub x, or the p... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So we have the constant on the outside, 2x times the derivative with respect to y, so it's 2x e to the y. Fair enough. Now what is the partial derivative of this with respect to x? So n sub x, or the partial of n with respect to x. So what's the derivative of sine of x with respect to x? Well, that's easy. That's cosin... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So n sub x, or the partial of n with respect to x. So what's the derivative of sine of x with respect to x? Well, that's easy. That's cosine of x. Plus 2x times e to the y, right? e to the y is just a constant because y is a constant when we're taking the partial with respect to x. So plus 2x e to the y. |
Exact equations example 1 First order differential equations Khan Academy.mp3 | That's cosine of x. Plus 2x times e to the y, right? e to the y is just a constant because y is a constant when we're taking the partial with respect to x. So plus 2x e to the y. And then minus 1, the derivative of a constant with respect to anything is going to be 0. So the derivative of n, the partial of n with respe... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So plus 2x e to the y. And then minus 1, the derivative of a constant with respect to anything is going to be 0. So the derivative of n, the partial of n with respect to x is cosine of x plus 2x e to the y, which lo and behold is the same thing as the derivative, the partial of m with respect to y. So there we have it.... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So there we have it. We've shown that m of y is equal to, or the partial of m with respect to y is equal to the partial of n with respect to x, which tells us that this is an exact equation. Now given that this is an exact equation, given that this is an exact equation. Oh, yeah, my wife snuck up behind me. I was wonde... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | Oh, yeah, my wife snuck up behind me. I was wondering whether I thought there was some critter in my house or something. Anyway, so we know that this is an exact equation. So what does that tell us? Well, that tells us that there's some xi where the partial derivative of xi with respect to x is equal to m, and the part... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So what does that tell us? Well, that tells us that there's some xi where the partial derivative of xi with respect to x is equal to m, and the partial derivative of xi with respect to y is equal to n. And if we know that xi, then we can rewrite our differential equation as the derivative of xi with respect to x is equ... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So we know that the partial of xi with respect to x is equal to m. So we could write that. We could write the partial of xi with respect to x is equal to m, which is y cosine of x plus 2xe to the y. That's just here. That's my m of x. We could have done it the other way. We could have said the partial of xi with respec... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | That's my m of x. We could have done it the other way. We could have said the partial of xi with respect to y is this thing over here. But let's just do it with x. Now, to at least get kind of a first approximation of what xi is, not an approximation, but to start to get a sense of it, let's take the derivative of both... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | But let's just do it with x. Now, to at least get kind of a first approximation of what xi is, not an approximation, but to start to get a sense of it, let's take the derivative of both sides with respect to sorry, take the anti-derivative, take the integral of both sides with respect to x. So if you take the derivativ... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So let me just write that down. The partial with respect to x, we're going to take the integrate with respect to x. That is going to be equal to the integral of this whole thing with respect to x. Cosine of x plus 2xe to the y. We're integrating with respect to x. And normally when you integrate with respect to x, you'... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | We're integrating with respect to x. And normally when you integrate with respect to x, you'd say, OK, plus c. But it actually could be a plus. Since this was a partial with respect to x, we could have had some function of y here in general. Because y, we treat it as a constant, right? And that makes sense. Because if ... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | Because y, we treat it as a constant, right? And that makes sense. Because if you were to take the partial of both sides of this with respect to x, if you were to take the partial of a function that is only a function of y with respect to x, you would have gotten a 0 here. So when you take the anti-derivative, we're li... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So when you take the anti-derivative, we're like, oh well, there might have been some function of y here that we lost when we took the partial with respect to x. So anyway, this will simplify to xi. Xi is going to be equal to the integral with respect to x, or the anti-derivative with respect to x here, plus some funct... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So let's do that. Let's figure out this integral. I'll do it in blue. So y is just a constant. So the anti-derivative y cosine of x is just y sine of x plus e to the y is constant. So 2x. The anti-derivative of 2x with respect to x is x squared. |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So y is just a constant. So the anti-derivative y cosine of x is just y sine of x plus e to the y is constant. So 2x. The anti-derivative of 2x with respect to x is x squared. So it's x squared e to the y. And then plus some function of y. And if you want to verify this is true, take the partial of this with respect to... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | The anti-derivative of 2x with respect to x is x squared. So it's x squared e to the y. And then plus some function of y. And if you want to verify this is true, take the partial of this with respect to x. If you take the partial of this with respect to x, you're going to get this in here, which is our function m up he... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | And if you want to verify this is true, take the partial of this with respect to x. If you take the partial of this with respect to x, you're going to get this in here, which is our function m up here, and then when you take the partial of this with respect to x, you'll get 0 and it'll get lost. OK. So we're almost the... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So we're almost there. We've almost figured out our xi, but we still need to figure out this function of y. Well, we know that if we take the partial of this with respect to y, since this is an exact equation, we should get this. We should get our n function. So let's do that. So the partial, I'll switch notation just ... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | We should get our n function. So let's do that. So the partial, I'll switch notation just to expose you to it. The partial of xi with respect to y is going to be equal to. So here, y sine of x, sine of x is just a constant, y is just y, so the derivative with respect to y is just sine of x. Plus, derivative of e to the... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | The partial of xi with respect to y is going to be equal to. So here, y sine of x, sine of x is just a constant, y is just y, so the derivative with respect to y is just sine of x. Plus, derivative of e to the y is e to the y, x squared is just a constant, so it's just x squared e to the y. Plus, what's the partial of ... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | Plus, what's the partial of f of y with respect to y? It's going to be f prime of y. And we also, so what did we do? We took m, we integrated with respect to x, and we said, well, we might have lost some function of y, so we added that to it. And then we took the partial of that xi that we've almost constructed, and we... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | We took m, we integrated with respect to x, and we said, well, we might have lost some function of y, so we added that to it. And then we took the partial of that xi that we've almost constructed, and we took the partial of that with respect to y. Now, we know, since this is exact, that that is going to equal our n. So... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | Cosine of x plus, so that's going to be equal to, I want to make sure I can read it up there, to our n, right? Oh no, sorry, n is up here. Sine of x, let me write that, sine of x plus x squared e to the y minus 1. So sine of x plus x squared e to the y minus 1. Plus x squared e to the y minus 1. That was just our n fro... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So sine of x plus x squared e to the y minus 1. Plus x squared e to the y minus 1. That was just our n from our original differential equation. And now we can solve for f prime of y. So let's see. We get sine of x plus x squared e to the y plus f prime of y is equal to sine of x plus x squared e to the y minus 1. So le... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | And now we can solve for f prime of y. So let's see. We get sine of x plus x squared e to the y plus f prime of y is equal to sine of x plus x squared e to the y minus 1. So let's see, we can delete sine of x from both sides. We can delete x squared e to the y from both sides. And then what are we left with? We're left... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So let's see, we can delete sine of x from both sides. We can delete x squared e to the y from both sides. And then what are we left with? We're left with f prime of y is equal to 1. And then we're left with f of y is equal to, well it equals y plus some constant c. So what is our xi now? We wrote our xi up here and we... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | We're left with f prime of y is equal to 1. And then we're left with f of y is equal to, well it equals y plus some constant c. So what is our xi now? We wrote our xi up here and we had this f of y here, so we can rewrite it now. So xi is a function of x and y. We've actually pretty much almost done solving it. Xi is a... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So xi is a function of x and y. We've actually pretty much almost done solving it. Xi is a function of x and y is equal to y sine of x plus x squared e to the y plus y. Oh sorry, this is f prime of y minus 1. So this is a minus 1. So this is a minus y plus c. So this is going to be a minus y plus c. So we solved for xi... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | So this is a minus 1. So this is a minus y plus c. So this is going to be a minus y plus c. So we solved for xi. And so what does that tell us? Well we said that original differential equation up here, using the partial derivative chain rule, that original differential equation, can be rewritten now as the derivative d... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | Well we said that original differential equation up here, using the partial derivative chain rule, that original differential equation, can be rewritten now as the derivative dx of xi is equal to, xi is a function of x and y, is equal to 0, or if you were to integrate both sides of this, you would get that xi of xy is ... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | Now we could say plus this c, plus this c, you call that c1 is equal to c2. Well you could subtract the c's from both sides and just be left with a c at the end. But anyway, we have solved this exact equation. One, first by recognizing it was exact, by taking the partial of this with respect to y and seeing if that was... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | One, first by recognizing it was exact, by taking the partial of this with respect to y and seeing if that was equal to the partial of n with respect to x. Once we saw that they were equal, we're like, OK, this is going to be exact. So let's figure out xi. Since this is exact, m is going to be the partial of xi with re... |
Exact equations example 1 First order differential equations Khan Academy.mp3 | Since this is exact, m is going to be the partial of xi with respect to x. n is the partial of xi with respect to y. Then to figure out y, we integrated m with respect to x, and we got this. But since we said, oh, well, instead of a plus c, it could have been a function of y there, because we took the partial with resp... |
Laplace transform of cos t and polynomials Laplace transform Khan Academy.mp3 | In the last video, we showed that the Laplace transform of f prime of t is equal to s times the Laplace transform of our function f minus f of 0. Now what we're going to do here is actually use this property that we showed is true and use it to fill in some more of the entries in our Laplace transform table that you'll... |
Laplace transform of cos t and polynomials Laplace transform Khan Academy.mp3 | So let's use these two things we know to figure out what the Laplace transform of cosine of at is. So the Laplace transform of cosine of at is equal to what? Well, if we assume that the Laplace transform of cosine of at is the derivative of some function, what is it the derivative of? If I were to, let me do it on the ... |
Laplace transform of cos t and polynomials Laplace transform Khan Academy.mp3 | If I were to, let me do it on the side. If f prime of t is equal to cosine of at, what is a potential f of t? What is a potential f of t? Well, it's the antiderivative, and we can just forget about the constant because we just have to know n f of t for which this is true. So what's the antiderivative of cosine of at? I... |
Laplace transform of cos t and polynomials Laplace transform Khan Academy.mp3 | Well, it's the antiderivative, and we can just forget about the constant because we just have to know n f of t for which this is true. So what's the antiderivative of cosine of at? It's 1 over a sine of at. So if this is f prime of t, then that is equal to s times the Laplace transform of its antiderivative, or 1 over ... |
Laplace transform of cos t and polynomials Laplace transform Khan Academy.mp3 | So if this is f prime of t, then that is equal to s times the Laplace transform of its antiderivative, or 1 over a sine of at minus the antiderivative evaluated at 0. Minus 1 over a sine of, well, a times 0 is 0. Well, sine of 0 is 0, so this whole term goes away. So this is equal to, well, this is a constant, right? T... |
Laplace transform of cos t and polynomials Laplace transform Khan Academy.mp3 | So this is equal to, well, this is a constant, right? This 1 over a, and we showed that the Laplace transform is a linear operator, so we can take it out. So this is equal to s over a times the Laplace transform of sine of at, and that is equal to s over a times a over f squared plus a squared, and the a's cancel out, ... |
Laplace transform of cos t and polynomials Laplace transform Khan Academy.mp3 | So then we get that the Laplace transform of cosine of at is equal to s over s squared plus a squared. And in three minutes, we filled in another table in our Laplace transform table. And now we have the two most important trig functions. Let's keep going. We haven't really done much with polynomials. We know a couple ... |
Laplace transform of cos t and polynomials Laplace transform Khan Academy.mp3 | Let's keep going. We haven't really done much with polynomials. We know a couple of things. We know that the Laplace transform of 1 is equal to 1 over s, so let's see if we could use this and the fact that the Laplace transform of f prime is equal to s times the Laplace transform of f minus f of 0. Or another way, let'... |
Laplace transform of cos t and polynomials Laplace transform Khan Academy.mp3 | We know that the Laplace transform of 1 is equal to 1 over s, so let's see if we could use this and the fact that the Laplace transform of f prime is equal to s times the Laplace transform of f minus f of 0. Or another way, let's rearrange this. Like if we know f, how can we figure out its Laplace transforms in terms o... |
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