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Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | When you give their product, you get two. And then when you add them, you get negative two. Well you can't have, or they could both be negative. But there's no two easy numbers, not one and two, none of those work. So if you can't factor this out right, the next idea is maybe you can complete the square and maybe this ... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | But there's no two easy numbers, not one and two, none of those work. So if you can't factor this out right, the next idea is maybe you can complete the square and maybe this will match one of the cosine or the sine formulas. So how can we complete the square in this denominator? Well this can be rewritten as, let me r... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | Well this can be rewritten as, let me rewrite this as s squared minus two s, and I'm gonna put a plus two out here. And this will hopefully, and you can watch my, I have a bunch of videos on completing the square if all of this looks foreign to you. And to complete the square, we just wanna turn this into a perfect squ... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | So to turn this into a perfect square, so something when I add to itself twice becomes minus two. And so that when I square it, when I add it to itself twice, it becomes minus two, it's minus one. And when I square, it'll become plus one. Plus one. I can't just add plus one arbitrarily to some expression, I have to mak... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | Plus one. I can't just add plus one arbitrarily to some expression, I have to make it neutral, so let me subtract one. I haven't changed this. I added one and I subtracted one. A little bit of a primer on completing the square. But by doing this, I now can call this expression right here I can now say that this thing i... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | I added one and I subtracted one. A little bit of a primer on completing the square. But by doing this, I now can call this expression right here I can now say that this thing is s minus one squared, and then this stuff out here, this out here is two minus one, this is just plus one. So I can rewrite my entire expressi... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | So I can rewrite my entire expression now as two times s minus one times e to the minus two s. Make sure I'm not clipping off at the top. E to the minus two s, all of that over s minus one squared plus one. So a couple of interesting things are starting to seem to be happening here. So if I just had, let's just do a co... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | So if I just had, let's just do a couple of test Laplace transforms. If I did the Laplace transform of cosine of t, we know that this is equal to s over s squared plus one. Which this kind of looks like. If this was an s and this was an s squared plus one. If this was f of s, then what is this? Well let's ignore this g... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | If this was an s and this was an s squared plus one. If this was f of s, then what is this? Well let's ignore this guy right here for a little bit. So what is, we know, actually from the last video, we saw well what if we took the Laplace transform, the Laplace transform of e to the, I'll call it one t, but let's say e... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | So what is, we know, actually from the last video, we saw well what if we took the Laplace transform, the Laplace transform of e to the, I'll call it one t, but let's say e to the, yeah let's write it, e to the one t times cosine of t. Well then this will just shift this Laplace transform by one. It'll shift it by one ... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | So this will be equal to s minus one over s minus one squared plus one. We're getting close, we're getting close. We now figured out this part right here. Now in the previous video, I think it was two videos ago, or maybe it was the last video, I forget. Memory fails me. I showed you that if you have the Laplace transf... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | Now in the previous video, I think it was two videos ago, or maybe it was the last video, I forget. Memory fails me. I showed you that if you have the Laplace transform of the unit step function of t times some f of t shifted by some value of c, then that this is equal to, this is equal to e to the minus cs times f of ... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | This can get very confusing, so I wanna be very careful here. Let's ignore all of this. I called this f of s before, but now I'm gonna backtrack a little bit. And let's just ignore this because I'm gonna redefine our f of s. Let's just ignore that for a second. Let's define our new f of t to be this. Let's say that tha... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | And let's just ignore this because I'm gonna redefine our f of s. Let's just ignore that for a second. Let's define our new f of t to be this. Let's say that that is f of t. Let's say f of t is equal to e to the t cosine of t. Then if you take the Laplace transform of that, that means that f of s is equal to s minus on... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | Nothing fancy there. I just defined our f of t as this, and then our f of s is that, right? Now, we have a situation here. This expression, let's ignore the two here. The two is just kind of a scaling factor. This expression right here, we can rewrite as that expression is equal to, if this is our f of s, this expressi... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | This expression, let's ignore the two here. The two is just kind of a scaling factor. This expression right here, we can rewrite as that expression is equal to, if this is our f of s, this expression right here is equal to two times our f of s times e to the minus two s. Or let me just write it. Let me switch the order... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | Let me switch the order just so we make it look right. Two times e to the minus two s times f of s. Well, that looks just like this if our two was equal to our c, right? If our two was equal to our c. So what does that tell us? That tells us that the inverse Laplace transform, if we take the inverse Laplace transform, ... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | That tells us that the inverse Laplace transform, if we take the inverse Laplace transform, and let's ignore the two. Well, we could just write, let's do the inverse Laplace transform of the whole thing. The inverse Laplace transform of this thing is going to be equal to, we can just write the two there as a scaling fa... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | Two there times this thing, times the unit step function. The unit step function, what's our c? If you just, you can just pattern match. You have a two here, you have a c, a minus c, a minus two. So c is two. The unit step function is zero until it gets to two times t, or of t, so, you know, and then it becomes one aft... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | You have a two here, you have a c, a minus c, a minus two. So c is two. The unit step function is zero until it gets to two times t, or of t, so, you know, and then it becomes one after t is equal to two, times our function shifted by two. By two. Shifted by two. So this is our inverse Laplace transform. Now, what was ... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | By two. Shifted by two. So this is our inverse Laplace transform. Now, what was our function? Our function was this thing right here. So if our inverse Laplace transform of that thing that I had written is this thing, and f of t, f of t is equal to e to the t cosine of t, then our inverse, let me write all of this down... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | Now, what was our function? Our function was this thing right here. So if our inverse Laplace transform of that thing that I had written is this thing, and f of t, f of t is equal to e to the t cosine of t, then our inverse, let me write all of this down. Let me write kind of our big result. We find we were established... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | Let me write kind of our big result. We find we were established that the inverse Laplace transform of that big thing that I had written before, two times s minus one, times e to the minus two, sorry, e to the minus two s, wouldn't have a t there if we, over s squared minus two s plus two is equal to this thing where f... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | I'll do it in another color just to ease the monotony. So it'd be e to the t minus two cosine of t minus two. Now you might be thinking, Sal, he must have taken all these baby steps with this problem because he's trying to explain it to me, but I'm taking baby steps with this problem so that I myself don't get confused... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | And let's just think about what baby steps we took, and I really want to review this. This is actually a surprisingly good problem. I didn't realize it when I first decided to do it. We saw this thing. We wanted to get this denominator into some form that is vaguely, vaguely useful to us, so I completed the square ther... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | We saw this thing. We wanted to get this denominator into some form that is vaguely, vaguely useful to us, so I completed the square there, and then we rewrote our Laplace transform, our f of s like this, and then we used a little pattern recognition. We said, look, if I take the Laplace transform of cosine of t, I'd g... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | It's s minus one over s minus one squared plus one. So we said, oh, well, that means that we're multiplying our original time domain function. We're multiplying our f of t times e to the one t, e to the one t, and that's what we got there, right? So the Laplace transform of e to the t cosine of t became s minus one ove... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | And if the roots of this characteristic equation are real, let's say we have two real roots, let me write that down, so the real scenario where the two solutions are going to be r1 and r2 where these are real numbers, then the general solution of this differential equation, and watch the previous videos if you don't re... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | What if they are complex? And just a little bit of review, what do I mean by that? Well if I wanted to figure out the roots of this and I didn't, you know, if I was lazy and I just wanted to do it without having to think, can I factor it, I would just immediately use the quadratic equation because that always works. An... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | And I would say, well the roots of my characteristic equation are negative b plus or minus the square root of b squared minus 4ac, all of that over 2a. So what do I mean by non-real roots? Well if this expression right here, if this b squared minus 4ac, if that's a negative number, then I'm going to have to take the sq... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | And so this whole term will actually become complex. We'll have a real part and an imaginary part. And actually the two roots are going to be conjugates of each other, right? We can rewrite this in the real and imaginary parts. We could rewrite this as the roots are going to be equal to minus b over 2a plus or minus th... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | We can rewrite this in the real and imaginary parts. We could rewrite this as the roots are going to be equal to minus b over 2a plus or minus the square root of b squared minus 4ac over 2a. And if b squared minus 4ac is less than 0, this is going to be an imaginary number. So in that case, let's just think about what ... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | So in that case, let's just think about what the roots look like generally and then we'll actually do some problems. So let me go back up here. So then the roots aren't going to be two real numbers like that. The roots, we can write them as two complex numbers that are conjugates of each other. And I think light blue i... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | The roots, we can write them as two complex numbers that are conjugates of each other. And I think light blue is a suitable color for that. So in that situation, let me write this as complex roots. This is a complex roots scenario. Then the roots of the characteristic equation are going to be some number. Let's call it... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | This is a complex roots scenario. Then the roots of the characteristic equation are going to be some number. Let's call it lambda. Let's call it mu. I think that's the convention that people use. Actually, let me see what they tend to use. It really doesn't matter. |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | Let's call it mu. I think that's the convention that people use. Actually, let me see what they tend to use. It really doesn't matter. Let's say it's lambda. So this number, some constant called lambda. And then plus or minus some imaginary number. |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | It really doesn't matter. Let's say it's lambda. So this number, some constant called lambda. And then plus or minus some imaginary number. And so it's going to be some constant mu. That's just some constant. I'm not trying to be fancy, but this is, I think, the convention used in most differential equations books. |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | And then plus or minus some imaginary number. And so it's going to be some constant mu. That's just some constant. I'm not trying to be fancy, but this is, I think, the convention used in most differential equations books. So it's mu times i. So these are the two roots. And these are two roots, right? |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | I'm not trying to be fancy, but this is, I think, the convention used in most differential equations books. So it's mu times i. So these are the two roots. And these are two roots, right? Because we have lambda plus mu i and lambda minus mu i. So these would be the two roots if b squared minus 4ac is less than 0. So le... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | And these are two roots, right? Because we have lambda plus mu i and lambda minus mu i. So these would be the two roots if b squared minus 4ac is less than 0. So let's see how that translates. Let's see what happens when we take these two roots and we put them into our general solution. So just like we'd learned before... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | So let's see how that translates. Let's see what happens when we take these two roots and we put them into our general solution. So just like we'd learned before, the general solution is going to be, I'll stay in the light blue, the general solution is going to be y is equal to c1 times e to the first root. Let's make ... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | Let's make that the plus version. So lambda plus mu i. All of that times x plus c2 times e to the second root. So that's going to be lambda minus mu i times x. Let's see if we can do some simplification here, because that i there really kind of makes things kind of crazy. So let's see if we can do anything to either ge... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | So that's going to be lambda minus mu i times x. Let's see if we can do some simplification here, because that i there really kind of makes things kind of crazy. So let's see if we can do anything to either get rid of it or simplify it, et cetera. So let's multiply the x out. Just do some algebraic manipulation. I'm tr... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | So let's multiply the x out. Just do some algebraic manipulation. I'm trying to use as much space as possible. So we get y is equal to c1 e to the what? Lambda x, just distributing that x, plus mu xi plus c2 times e to the lambda x minus mu xi. Just distributed the x's in both of the terms. And let's see what we can do... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | So we get y is equal to c1 e to the what? Lambda x, just distributing that x, plus mu xi plus c2 times e to the lambda x minus mu xi. Just distributed the x's in both of the terms. And let's see what we can do. Well, when you add exponents, this is the exact same thing as y is equal to c1 e to the lambda x times e to t... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | And let's see what we can do. Well, when you add exponents, this is the exact same thing as y is equal to c1 e to the lambda x times e to the mu xi. If you have the same base and you're multiplying, you could just add exponents. So this is the same thing as that. Plus c2 times e to the lambda x times e to the minus mu ... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | So this is the same thing as that. Plus c2 times e to the lambda x times e to the minus mu xi. And let's see, we have an e to the lambda x in both of these terms, so we can factor it out. So we get y is equal to, let me draw a line here. I don't want you to get confused with all this quadratic equation stuff. y is equa... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | So we get y is equal to, let me draw a line here. I don't want you to get confused with all this quadratic equation stuff. y is equal to e to the lambda x times c1 e to the mu xi, that's an i, plus c2 times e to the minus mu xi. Now what can we do? And this is where it gets fun. If you watched the calculus playlist, es... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | Now what can we do? And this is where it gets fun. If you watched the calculus playlist, especially when I talk about approximating functions with series, we came up with what I thought was the most amazing result in calculus, or in mathematics, just from a metaphysical point of view. And now we will actually use it fo... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | And now we will actually use it for something that you'll hopefully see is vaguely useful. So here we have two terms that have something times e to the something times i. And we learned before Euler's formula. And what was Euler's formula? I'll write that in purple. That e to the i theta, or we could write e to the ix,... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | And what was Euler's formula? I'll write that in purple. That e to the i theta, or we could write e to the ix, is equal to cosine of x plus i sine of x. And what's amazing about that is if you put negative 1 in here, then you get e to the i pi is equal to negative 1. If you substitute this, cosine of pi is 0. So I thou... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | And what's amazing about that is if you put negative 1 in here, then you get e to the i pi is equal to negative 1. If you substitute this, cosine of pi is 0. So I thought that was amazing. Or you could write e to the i 2 pi is equal to 1. That's pretty amazing as well. So in one equation you have all of the fundamental... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | Or you could write e to the i 2 pi is equal to 1. That's pretty amazing as well. So in one equation you have all of the fundamental numbers of mathematics. So that's amazing. But let's get back down to earth and get practical. So let's see if we can use this to simplify Euler's. This is actually a definition. |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | So that's amazing. But let's get back down to earth and get practical. So let's see if we can use this to simplify Euler's. This is actually a definition. And the definition makes a lot of sense. Because when you do the power series approximation, or the Maclaurin series approximation of e to the x, it really looks lik... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | This is actually a definition. And the definition makes a lot of sense. Because when you do the power series approximation, or the Maclaurin series approximation of e to the x, it really looks like, or e to the i, right, e to the x, it really does look like cosine of x plus i times the power series approximation of x. ... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | But anyway, we won't go into that now. I have like six or seven videos on it. But let's use this to simplify this up here. So we can rewrite that as y is equal to e to the lambda x times, let's do the first one, c1. It's e to the mu x i. So that can be rewritten, e to the mu x i, so instead of an x, we have a mu x, tha... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | So we can rewrite that as y is equal to e to the lambda x times, let's do the first one, c1. It's e to the mu x i. So that can be rewritten, e to the mu x i, so instead of an x, we have a mu x, that will be equal to cosine of whatever is in front of the i. So cosine of mu x plus i sine of mu x. And then plus c2. Times ... |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | So cosine of mu x plus i sine of mu x. And then plus c2. Times what? Times cosine of minus mu x plus i sine of minus mu x. And let's see if we can simplify this further. So one thing that you might want, let's distribute the c's. So now we get, I'll do it a different color. |
Complex roots of the characteristic equations 1 Second order differential equations Khan Academy.mp3 | Times cosine of minus mu x plus i sine of minus mu x. And let's see if we can simplify this further. So one thing that you might want, let's distribute the c's. So now we get, I'll do it a different color. Actually, I'm running out of time. So I'll continue this in the next video. See you soon. |
Modeling population with simple differential equation Khan Academy.mp3 | Modeling population. We're actually gonna go into some depth on this eventually, but here we're gonna start with simpler models. And we'll see, we will stumble on, using the logic of differential equations, things that you might have seen in your algebra or your pre-calculus class. So on some level, what we're going to... |
Modeling population with simple differential equation Khan Academy.mp3 | So on some level, what we're going to do here is going to be review, but we're gonna get there using the power of modeling with differential equations. So let's just define some variables. Let's say that P is equal to our population. And let's say that T is, let's say that T is equal to the time that has passed in days... |
Modeling population with simple differential equation Khan Academy.mp3 | And let's say that T is, let's say that T is equal to the time that has passed in days. In days. It could have been years or months. But let's say we're doing the population of insects that reproduce quite quickly. So days seem like a nice time span to care about. Now, what would be a reasonable model? Well, we could s... |
Modeling population with simple differential equation Khan Academy.mp3 | But let's say we're doing the population of insects that reproduce quite quickly. So days seem like a nice time span to care about. Now, what would be a reasonable model? Well, we could say that the rate of change, the rate of change of our population with respect to time, with respect to time, is, well, a reasonable t... |
Modeling population with simple differential equation Khan Academy.mp3 | Well, we could say that the rate of change, the rate of change of our population with respect to time, with respect to time, is, well, a reasonable thing to say is that it's going to be proportional to the actual population. The actual population. Why is that reasonable? Well, the larger the population, the larger the ... |
Modeling population with simple differential equation Khan Academy.mp3 | Well, the larger the population, the larger the rate at any given time. If you have a thousand people, the rate at which they're reproducing is going to be more, or a thousand insects, is going to be more insects per second or per day or per year than if you only have 10 insects. So it makes sense that the rate of grow... |
Modeling population with simple differential equation Khan Academy.mp3 | And so, you know, sometimes you think of differential equations as these daunting, complex things, but notice, we've just been able to express a reasonably not-so-complicated idea. The rate of change of population is going to be proportional to the population. And now, once we've expressed that, we can actually try to ... |
Modeling population with simple differential equation Khan Academy.mp3 | So how do we do that? And I encourage you to pause the video at any time and see if you can solve this differential equation. So assuming you at least maybe have had an attempt at it. And you might immediately recognize that this is a separable differential equation. And in separable differential equations, we want one... |
Modeling population with simple differential equation Khan Academy.mp3 | And you might immediately recognize that this is a separable differential equation. And in separable differential equations, we want one variable and all the differentials involving that variable on one side, and the other variable and all the differentials involving the other variable on the other side, and then we ca... |
Modeling population with simple differential equation Khan Academy.mp3 | This is the limit as our change in P over change in time. This is our instantaneous change. But for the sake of separable differential equations or differential equations in general, you can treat this derivative in Leibniz notations like fractions, and you can treat these differentials like quantities because we will ... |
Modeling population with simple differential equation Khan Academy.mp3 | So let's do that. So we want to put all the P's and dP's on one side, and all the things that involve t or that I guess don't involve P on the other side. So we could divide both sides by P. We could divide both sides by P. And so we'll have one over P. You have one over P here, and then those will cancel. And then you... |
Modeling population with simple differential equation Khan Academy.mp3 | And then you can multiply both sides times dt. We could multiply both sides times dt. Once again, treating the differential like a quantity, which it really isn't a quantity. You really have to view this as a limit of that change in P over change in time. The limit as we get smaller and smaller and smaller changes in t... |
Modeling population with simple differential equation Khan Academy.mp3 | You really have to view this as a limit of that change in P over change in time. The limit as we get smaller and smaller and smaller changes in time. But once again, for the sake of this, we can do this. And when we do that, we would be left with one over P dP is equal to K dt. Now we can integrate both sides. Because ... |
Modeling population with simple differential equation Khan Academy.mp3 | And when we do that, we would be left with one over P dP is equal to K dt. Now we can integrate both sides. Because this was a separable differential equation, we were able to completely separate the P's and dP's from the things involving T's, or I guess the things that aren't involving P's. And then if we integrate th... |
Modeling population with simple differential equation Khan Academy.mp3 | And then if we integrate this side, we would get the natural log of the absolute value of our population. And we could say plus some constant if we want, but we're gonna get a constant on this side as well. So we could just say that's going to be equal to K times T plus some constant. I'll just call that C1. And once a... |
Modeling population with simple differential equation Khan Academy.mp3 | I'll just call that C1. And once again, I could have put a plus C2 here, but I could have then subtracted the constant from both sides, and I would just get the constant on the right-hand side. Now, how can I solve for P? Well, the natural log of the absolute value of P is equal to this thing right over here. That mean... |
Modeling population with simple differential equation Khan Academy.mp3 | Well, the natural log of the absolute value of P is equal to this thing right over here. That means, that's the same thing, that means that the absolute value of P is equal to E to all of this business. E to the, let me do the same colors, KT plus C1. Now, this right over here is the same thing, just using our exponent... |
Modeling population with simple differential equation Khan Academy.mp3 | Now, this right over here is the same thing, just using our exponent properties, this is the same thing as E to the KT, AE to the K times T times E to the C1. Now, this is just E to some constant, so we could just call this, let's just call that the constant C. So, this is all simplified to CE, CE to the KT, to the KT.... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | This is a separable differential equation, and we can integrate things quite easily. But as you will see as you go further in the world of differential equations, most differential equations are not so easy to solve. In fact, many of them are impossible to solve using analytic methods. And so given that, what do you do... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | And so given that, what do you do? You've nicely described some phenomena, modeled some phenomena using differential equations, but if you can't solve it analytically, do you just give up? And the answer to that question is no. You do not just give up, because we now have computers, and computers are really good at num... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | You do not just give up, because we now have computers, and computers are really good at numerical methods, numerical methods for approximating and giving us a sense of what the solution to a differential equation might look like. And so how do we do that? Well, in this video, we can explore one of the most straightfor... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | So what we do is, so let me draw a little table here. So a little table here. And so, actually let me give myself some, I'm gonna do it over here on the left-hand side. So a little table. So x and then y, with x, y, and then dy, dx. And you could set up a table like this to create a slope field. You could just pick all... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | So a little table. So x and then y, with x, y, and then dy, dx. And you could set up a table like this to create a slope field. You could just pick all the, you could sample x's and y's in the xy plane, and then figure out, for a first-order differential equation like this, what is the slope going to be at that point, ... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | You could just pick all the, you could sample x's and y's in the xy plane, and then figure out, for a first-order differential equation like this, what is the slope going to be at that point, and you could construct a slope field. And we're gonna do something kind of related, but instead of trying to construct a slope ... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | We know that the particular solution of this differential equation contains this point. So we're gonna start with that point. So we're gonna start with x is equal to zero, and let me do this in a different color. We're gonna start with x is equal to zero, y is equal to one, which is that point right over there. And we'... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | We're gonna start with x is equal to zero, y is equal to one, which is that point right over there. And we're gonna say, well, okay, what is the derivative at that point? Well, we know the derivative at any point or any solution to this differential equation, the derivative is going to be equal to the y value. So in th... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | So in this case, the derivative is going to be equal to y. It's going to be equal to one. And in general, if the derivative, just like what we saw in the case of slope fields, as long as the derivative is expressed as a function of x's and y of x's, then you can figure out what the slope of the tangent line will be at ... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | And so you say, okay, there's a slope of one at that point. So I can depict it like that. And instead of just keep doing that at a bunch of points, you say, okay, well, let's just, we know that the slope is changing, or it's probably changing for most cases, but let's just assume it's fixed until our next x, and then u... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | So what am I talking about here? So when I talk about the next x, we're talking about, well, let's just step, let's say for the sake of simplicity, we're gonna have a delta x of one, a change in x of one. So we're gonna step from x equals zero now. We're gonna now step from that to x is equal to one. So we're now gonna... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | We're gonna now step from that to x is equal to one. So we're now gonna go to, actually, let me not use that, I used that yellow color already for the actual graph, or for the actual e to the x. So now let's say x is equal to one. So we've, our delta x is one. So we've just added one here. And what we can do in our lit... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | So we've, our delta x is one. So we've just added one here. And what we can do in our little approximation scheme here is let's just assume that that slope was constant over that interval. So where does that get us to? Well, if y was at one, and if I have a slope of one, for one more, for one increase in x, I'm gonna i... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | So where does that get us to? Well, if y was at one, and if I have a slope of one, for one more, for one increase in x, I'm gonna increase by y by one. So then y is going to increase by one, and is going to get to two. And we see that point right over there. And you already might see where this is going. Now, if this w... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | And we see that point right over there. And you already might see where this is going. Now, if this were actually a point on the curve, on the solution, and if it was satisfying this, what would then the derivative be? Well, the derivative is equal to y. The slope of the tangent line is going to be equal to y. So in th... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | Well, the derivative is equal to y. The slope of the tangent line is going to be equal to y. So in this case, the slope of the tangent line is now going to be equal to two. And we could depict that, let me depict that in magenta here. So it is going to be, it is going to be two. So it's gonna look, the slope of the tan... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | And we could depict that, let me depict that in magenta here. So it is going to be, it is going to be two. So it's gonna look, the slope of the tangent line there is going to be two. And so what does that tell us? Well, if we step, if we step by our delta x one more, so now our x is equal to two, what should the corres... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | And so what does that tell us? Well, if we step, if we step by our delta x one more, so now our x is equal to two, what should the corresponding, what should the corresponding y be? Well, let's see. Now, for every one that we increase in the x direction, we should increase two in the y direction, because the slope is t... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | Now, for every one that we increase in the x direction, we should increase two in the y direction, because the slope is two. So the very next one should be four. Y is equal to four. So we could imagine, we have now kind of had a constant slope, and we get to that point right over there. And now we can do the same thing... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | So we could imagine, we have now kind of had a constant slope, and we get to that point right over there. And now we can do the same thing. Well, if we assume dy dx, based on the differential equation, has to be equal to y, we say, okay, the slope of the tangent line there is going to be the same thing as y. It's going... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | It's going to be four. And so if we step our x up by one, if we increment our x by one again, and once again, we just decided to increment by one. We could have incremented by 10. We could have incremented by.01. And you could guess which one's going to give you a more accurate result. But if we step up by one now, and... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | We could have incremented by.01. And you could guess which one's going to give you a more accurate result. But if we step up by one now, and our slope is four, well, we're going to increase by, if we increased x by one, we're going to increase y by four. So we are going to get, we are going to get to eight. And so we a... |
Euler's method Differential equations AP Calculus BC Khan Academy.mp3 | So we are going to get, we are going to get to eight. And so we are at the.3,8, which is right over here. And so for this next stretch, the next stretch is going to look like that. And as you can see, just by doing this, we have been able to approximate what the particular solution looks like. And you might say, hey, S... |
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