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Complex roots of the characteristic equations 2 Second order differential equations Khan Academy.mp3 | And the last thing we can simplify is, well, you know, c1 and c2 are arbitrary constants. So let's just define this as another constant. I don't know, let's call it, I'll just call it c3, I mean, just to not confuse you by using c1 twice. I'll call this c3. And now this might be a little bit of a stretch for you, but i... |
Complex roots of the characteristic equations 2 Second order differential equations Khan Academy.mp3 | I'll call this c3. And now this might be a little bit of a stretch for you, but if you think about it, it really makes sense. This is still just a constant, right? Especially if I say, you know what, I'm not restricting the constants to the reals, c could be an imaginary number. So if c is an imaginary number or some t... |
Complex roots of the characteristic equations 2 Second order differential equations Khan Academy.mp3 | Especially if I say, you know what, I'm not restricting the constants to the reals, c could be an imaginary number. So if c is an imaginary number or some type of complex number, we don't even know whether this is necessarily an imaginary number. So we're not going to make any assumptions about it. Let's just say that ... |
Complex roots of the characteristic equations 2 Second order differential equations Khan Academy.mp3 | Let's just say that this is some other arbitrary constant. Call this c4, and we can worry about it when we're actually given the initial conditions. But what this gives us, if we make that simplification, we actually get a pretty straightforward general solution to our differential equation where the characteristic equ... |
Complex roots of the characteristic equations 2 Second order differential equations Khan Academy.mp3 | And that I'll do in a new color. That is, y is equal to e to the lambda x times some constant, I'll call it c3, it could be c1, it could be c100, whatever, some constant times cosine of mu of x plus some other constant, I called it c4, it doesn't have to be c4, I just didn't want to confuse it with these, but some othe... |
Complex roots of the characteristic equations 2 Second order differential equations Khan Academy.mp3 | One is, we haven't done anything different. At the end of the day, we still just took the two roots and substituted it back into these equations for r1 and r2. The difference is, we just kept algebraically simplifying it so that we got rid of the i's. That's all we did. There was really nothing new here except for some... |
Complex roots of the characteristic equations 2 Second order differential equations Khan Academy.mp3 | That's all we did. There was really nothing new here except for some algebra and the use of Euler's formula. But when r1 and r2 involved complex numbers, we got to this simplification. So in general, if you get the characteristic equation and your two roots are lambda plus or minus mu i, then the general solution is go... |
Complex roots of the characteristic equations 2 Second order differential equations Khan Academy.mp3 | So in general, if you get the characteristic equation and your two roots are lambda plus or minus mu i, then the general solution is going to be this. And it's not too, if you had to memorize it, although I don't want you to, you should be able to derive this on your own. But it's not too hard to think. And actually, i... |
Complex roots of the characteristic equations 2 Second order differential equations Khan Academy.mp3 | And actually, if you ever forget it, just solve your characteristic equation, get your complex numbers, and just substitute it right back in this equation. And then with the real numbers, instead of the lambda and the mu, with the real numbers, just do the simplification we did and you'll get to the exact same point. B... |
Complex roots of the characteristic equations 2 Second order differential equations Khan Academy.mp3 | And let's see if we can do a problem real fast that involves that. So let's say I had the differential equation y prime prime plus the first derivative plus y is equal to 0, so our characteristic equation is r squared plus r plus 1 is equal to 0. Let's break out the quadratic formula. So the roots are going to be negat... |
Complex roots of the characteristic equations 2 Second order differential equations Khan Academy.mp3 | So the roots are going to be negative b, so it's negative 1 plus or minus the square root of b squared, b squared is 1, minus 4 times ac. Well, a and c are both 1, so it's just minus 4. All of that over 2, right? 2 times a, all of that over 2. So the roots are going to be negative 1 plus or minus the square root of, th... |
Complex roots of the characteristic equations 2 Second order differential equations Khan Academy.mp3 | 2 times a, all of that over 2. So the roots are going to be negative 1 plus or minus the square root of, this is negative 3, over 2. Or we could rewrite this as the roots are r is equal to negative 1 half plus or minus, well, we could rewrite this as i times the square root of 3, or square root of 3i, over 2. Or we cou... |
Complex roots of the characteristic equations 2 Second order differential equations Khan Academy.mp3 | Or we could write this as square root of 3 over 2 times i, actually that's the best way to write it. You just take the i out, so it takes the negative 1 out, and you have left with square root of 3 over 2. So these are the roots, and now if we want the general solution, we just have to throw this right back into that, ... |
Complex roots of the characteristic equations 2 Second order differential equations Khan Academy.mp3 | So let me write that right down here. So our general solution will be y is equal to e to the real part of our complex conjugate, so e to the minus 1 half times x, this is our lambda, times some constant, I'll write c1 now. c1 times cosine of the imaginary part without the i, so cosine of square root of 3 over 2x plus c... |
Particular solution to differential equation example Khan Academy.mp3 | So let's say I have the differential equation, the derivative of y with respect to x is equal to two y squared. And let's say that the graph of a particular solution to this the graph of a particular solution passes through the point one comma negative one. So my question to you is, what is y, what is y when x is equal... |
Particular solution to differential equation example Khan Academy.mp3 | So the particular solution to the differential equation that passes through the point one comma negative one, what is y when x is equal to three? And I encourage you to pause the video and try to work through it on your own. So I'm assuming you had a go at it and the key with a separable differential equation, and that... |
Particular solution to differential equation example Khan Academy.mp3 | So how do you do that here? Well, what I could do, let me just rewrite it. So it's gonna be dy dx is equal to two y squared. Is equal to two y, equal to two y squared. So let's see, we can multiply both sides by dx. And let's see, so then we're gonna have, that cancels with that if we treat it as just a value or as a v... |
Particular solution to differential equation example Khan Academy.mp3 | Is equal to two y, equal to two y squared. So let's see, we can multiply both sides by dx. And let's see, so then we're gonna have, that cancels with that if we treat it as just a value or as a variable, we're gonna have dy is equal to two y squared dx. Well, we're not quite done yet. We need to get this two y squared ... |
Particular solution to differential equation example Khan Academy.mp3 | Well, we're not quite done yet. We need to get this two y squared on the left-hand side. So we can divide both sides by two y squared. So if we divide both sides by two y squared, two y squared, the left-hand side, we could rewrite this as 1 1 2 y to the negative two power is going to be equal to dy. Let me, dy is equa... |
Particular solution to differential equation example Khan Academy.mp3 | So if we divide both sides by two y squared, two y squared, the left-hand side, we could rewrite this as 1 1 2 y to the negative two power is going to be equal to dy. Let me, dy is equal to dx. And now we can integrate both sides. So we can integrate both sides. Let me get myself a little bit more space. And so what is... |
Particular solution to differential equation example Khan Academy.mp3 | So we can integrate both sides. Let me get myself a little bit more space. And so what is, what is this left-hand side going to be? Well, we increment the exponent and then divide by that value. So y to the negative two, if you increment it to y to the negative one and then divide by negative one, so this is going to b... |
Particular solution to differential equation example Khan Academy.mp3 | Well, we increment the exponent and then divide by that value. So y to the negative two, if you increment it to y to the negative one and then divide by negative one, so this is going to be negative 1 1 2 y to the negative one power. And we could do a plus c like we did in the previous video, but we're gonna have a plu... |
Particular solution to differential equation example Khan Academy.mp3 | And you could subtract, or you know, you have different arbitrary constants on both sides and you could subtract them from each other. So I'm just gonna write the constant only on one side. So you have that is equal to, well, if I integrate just dx, that's just going to give me x. That's just gonna give me x. So this r... |
Particular solution to differential equation example Khan Academy.mp3 | That's just gonna give me x. So this right over here is x, and of course I can have a plus c over there. And if I want, I can solve for y. If I multiply, let's see, I can multiply both sides by negative two, and then I'm gonna have, the left-hand side you're just gonna have y to the negative one, or one over y. Is equa... |
Particular solution to differential equation example Khan Academy.mp3 | If I multiply, let's see, I can multiply both sides by negative two, and then I'm gonna have, the left-hand side you're just gonna have y to the negative one, or one over y. Is equal to, if I multiply the right-hand side times negative two, I'm gonna have negative two times x plus, well, it's some arbitrary constant. I... |
Particular solution to differential equation example Khan Academy.mp3 | And then if we want, we can take the reciprocal of both sides. And so we will get y is equal to, is equal to one over negative two x plus c. And now we can use, we can use the information they gave us right over here, the fact that our particular solution needs to go through this point to solve for c. So when x is nega... |
Particular solution to differential equation example Khan Academy.mp3 | We could multiply both sides times c minus two. If then we will get, actually let me just scroll down a little bit. So if you multiply both sides times c minus two, negative one times c minus two is gonna be negative c plus two, or two minus c is equal to one. All I did is I multiplied c minus two times both sides. And... |
Particular solution to differential equation example Khan Academy.mp3 | All I did is I multiplied c minus two times both sides. And then let's see, I can subtract two from both sides. So negative c is equal to negative one. And then if I multiply both sides by negative one, we get c is equal to one. So our particular solution is y is equal to one over negative two x plus one. And we are al... |
Particular solution to differential equation example Khan Academy.mp3 | And then if I multiply both sides by negative one, we get c is equal to one. So our particular solution is y is equal to one over negative two x plus one. And we are almost done. They didn't just ask for, we didn't just ask for the particular solution. We asked what is y when x is equal to three. So y is going to be eq... |
Worked example slope field from equation AP Calculus AB Khan Academy.mp3 | Well, the easiest way to think about a slope field, if I needed to plot this slope field by hand, I would sample a bunch of x and y points and then I would figure out what the derivative would have to be at that point. And so what we can do here, since they've already drawn some candidate slope fields for us, is figure... |
Worked example slope field from equation AP Calculus AB Khan Academy.mp3 | So I'm gonna have, I'm gonna have x, y, and then the derivative of y with respect to x. And we can do it at a bunch of values, so let's think about it. Let's think about when, we're at this point right over here, when x is two and y is two. When x is two and y is two, the derivative of y with respect to x is going to b... |
Worked example slope field from equation AP Calculus AB Khan Academy.mp3 | When x is two and y is two, the derivative of y with respect to x is going to be two minus two. It's going to be equal to zero. And just with that, let's see, here, this slope on this slope field does not look like it's zero. This looks like it's negative one. So already I could rule this one out. This slope right over... |
Worked example slope field from equation AP Calculus AB Khan Academy.mp3 | This looks like it's negative one. So already I could rule this one out. This slope right over here looks like it's positive one, so I'll rule that out. It's definitely not zero. This slope also looks like positive one, so I can rule that one out. This slope at two comma two actually does look like zero, so I'm liking ... |
Worked example slope field from equation AP Calculus AB Khan Academy.mp3 | It's definitely not zero. This slope also looks like positive one, so I can rule that one out. This slope at two comma two actually does look like zero, so I'm liking this one right over here. This slope at two comma two looks larger than one, so I could rule that out. So it was that straightforward to deduce that this... |
Worked example slope field from equation AP Calculus AB Khan Academy.mp3 | This slope at two comma two looks larger than one, so I could rule that out. So it was that straightforward to deduce that this choice right over here is, if any of these are going to be the accurate slope field, it's this one. But just for kicks, we could keep going to verify that this is indeed the slope field. So le... |
Worked example slope field from equation AP Calculus AB Khan Academy.mp3 | So let's think about what happens when x is equal to, well, one, whenever x is equal to y, you're going to get the derivative equaling zero, and you see that here. When you're at four, four, derivative equals zero. When it's six, six, derivative equals zero. At negative two, negative two, derivative equals zero. So tha... |
Worked example slope field from equation AP Calculus AB Khan Academy.mp3 | At negative two, negative two, derivative equals zero. So that feels good that this is the right slope field. And then we could pick other arbitrary points. Let's say when x is four, y is two, then the derivative here should be four minus two, which is going to be two. So when x is four, y is two, we do indeed see that... |
Worked example slope field from equation AP Calculus AB Khan Academy.mp3 | Let's say when x is four, y is two, then the derivative here should be four minus two, which is going to be two. So when x is four, y is two, we do indeed see that the slope field is indicating a slope that looks like two right over here. And if it was the other way around, when x is, when x is, let's say, x is negativ... |
Worked example slope field from equation AP Calculus AB Khan Academy.mp3 | So negative four, negative two. Well, negative four minus negative two is going to be negative two. And you can see that right over here. Negative four, negative two. You can see the slope right over here. It's a little harder to see. Looks like negative two. |
Writing a differential equation Differential equations AP Calculus AB Khan Academy.mp3 | A particle moves along a straight line. Its speed is inversely proportional to the square of the distance s it has traveled. Which equation describes this relationship? So I'm not gonna even look at these choices, and I'm just gonna try to parse this sentence up here and see if we can come up with an equation. So they ... |
Writing a differential equation Differential equations AP Calculus AB Khan Academy.mp3 | So I'm not gonna even look at these choices, and I'm just gonna try to parse this sentence up here and see if we can come up with an equation. So they tell us its speed is inversely proportional to what? To the square of the distance s it has traveled. So s is equal to distance. S is equal to distance. And how would we... |
Writing a differential equation Differential equations AP Calculus AB Khan Academy.mp3 | So s is equal to distance. S is equal to distance. And how would we denote speed then, if s is distance? Well, speed is the rate of change of distance with respect to time. So our speed would be the rate of distance with respect to time. The rate of change of distance with respect to time. So this is going to be our sp... |
Writing a differential equation Differential equations AP Calculus AB Khan Academy.mp3 | Well, speed is the rate of change of distance with respect to time. So our speed would be the rate of distance with respect to time. The rate of change of distance with respect to time. So this is going to be our speed. So now that we got our notation, the s is the distance, the derivative of s with respect to time is ... |
Writing a differential equation Differential equations AP Calculus AB Khan Academy.mp3 | So this is going to be our speed. So now that we got our notation, the s is the distance, the derivative of s with respect to time is speed, we can say the speed, which is d capital S, dt, is inversely proportional. So it's inversely proportional. I'll write a proportionality constant over what? It's inversely proporti... |
Writing a differential equation Differential equations AP Calculus AB Khan Academy.mp3 | I'll write a proportionality constant over what? It's inversely proportional to what? To the square of the distance. To the square of the distance it has traveled. So there you go. This is an equation that I think is describing a differential equation, really, that's describing what we have up here. Now let's see which... |
Writing a differential equation Differential equations AP Calculus AB Khan Academy.mp3 | To the square of the distance it has traveled. So there you go. This is an equation that I think is describing a differential equation, really, that's describing what we have up here. Now let's see which of these choices match that. Well, actually, this one is exactly what we wrote. The speed, the rate of change of dis... |
Writing a differential equation Differential equations AP Calculus AB Khan Academy.mp3 | Now let's see which of these choices match that. Well, actually, this one is exactly what we wrote. The speed, the rate of change of distance with respect to time is inversely proportional to the square of the distance. Now just to make sure we understand these other ones, let's just interpret them. This is saying that... |
Writing a differential equation Differential equations AP Calculus AB Khan Academy.mp3 | Now just to make sure we understand these other ones, let's just interpret them. This is saying that the distance, which is a function of time, is inversely proportional to the time squared. That's not what they told us. This is saying that the distance is inversely proportional to the distance squared. That one is esp... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | So now that we've spent some time thinking about what a differential equation is, and even visualizing solutions to a differential equation using things like slope field, let's start seeing if we can actually solve differential equations. And as we'll see, different types of differential equations might require differe... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | But let's go to what I would argue is the simplest form of differential equation to solve, and that's what's called a separable, separable differential equation. And we will see in a second why it is called a separable differential equation. So let's say that we have the derivative of y with respect to x is equal to ne... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | So we have this differential equation, and we want to find the particular solution that goes through the point, that goes through the point zero comma one. And I encourage you to pause this video, and I'll give you a hint. If you can, on one side of this equation, through algebra, separate out the y's and the d y's, an... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | Now if you can't do it, don't worry, because we're about to work through it. So as I said, let's use a little bit of algebra to get all the y's and d y's on one side, and all the x's and d x's on the other side. So one way, let's say I want to get all the y's and d y's on the left hand side, and all the x's and d x's o... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | Well I can multiply both sides times y. So I can multiply both sides times y. That has the effect of putting the y's on the left hand side. And then I can multiply both sides times d x. I can multiply both sides times d x. And we kind of treat, you can treat these differentials as you would treat a variable when you're... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | And then I can multiply both sides times d x. I can multiply both sides times d x. And we kind of treat, you can treat these differentials as you would treat a variable when you're manipulating it to essentially separate out the variables. And so this will cancel with that. And so we are left with, we are left with y d... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | And so we are left with, we are left with y d y, y d y is equal to negative x, and actually let me write it this way. Let me write it as negative x e, actually I might want a little more space. So negative x e to the negative x squared d x. D x. Now why is this interesting? Because we can integrate both sides. And now ... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | D x. Now why is this interesting? Because we can integrate both sides. And now this also highlights why we call this separable. You won't be able to do this with every differential equation. You won't be able to algebraically separate the y's and d y's on one side, and the x's and d x's on the other side. But this one ... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | And now this also highlights why we call this separable. You won't be able to do this with every differential equation. You won't be able to algebraically separate the y's and d y's on one side, and the x's and d x's on the other side. But this one we were able to. And so that's why this is called a separable different... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | But this one we were able to. And so that's why this is called a separable differential equation. Differential, differential equation. And it's usually the first technique that you should try. Hey, can I separate the y's and the x's? And as I said, this is not going to be true of many, if not most differential equation... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | And it's usually the first technique that you should try. Hey, can I separate the y's and the x's? And as I said, this is not going to be true of many, if not most differential equations. But now that we did this, we can integrate both sides. So let's do that. So I'll find a nice color to integrate with. So I'm going t... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | But now that we did this, we can integrate both sides. So let's do that. So I'll find a nice color to integrate with. So I'm going to integrate, integrate both sides. Now if you integrate the left-hand side, what do you get? You get, and remember, we're integrating with respect to y here. So this is going to be y squar... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | So I'm going to integrate, integrate both sides. Now if you integrate the left-hand side, what do you get? You get, and remember, we're integrating with respect to y here. So this is going to be y squared over two. And we could put some constant there. I could call that plus c one. And if you're integrating, now that's... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | So this is going to be y squared over two. And we could put some constant there. I could call that plus c one. And if you're integrating, now that's going to be equal to, now the right-hand side we're integrating with respect to x. And let's see, you could do u substitution, or you could recognize that look, the deriva... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | And if you're integrating, now that's going to be equal to, now the right-hand side we're integrating with respect to x. And let's see, you could do u substitution, or you could recognize that look, the derivative of negative x squared is going to be negative two x. So if that was a two there, and if you don't want to ... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | So now you could either do u substitution explicitly, or you could do it in your head, where you said u is equal to negative x squared, and then du will be negative two x dx, or you can kind of do this in your head at this point. So I have something and its derivative, so I really could just integrate with respect to t... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | The antiderivative of this is e to the negative x squared, and then of course I might have some other constant. I'll just call that c two. And once again, if this part over here, what I just did seems strange, the u substitution, you might want to review that piece. Now, what can I do here? Well I have a constant on th... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | Now, what can I do here? Well I have a constant on the left-hand side, it's an arbitrary constant, we don't know what it is. I haven't used this initial condition yet, we could call it. So let me just subtract c one from both sides. So if I just subtract c one from both sides, I have an arbitrary, so this is going to c... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | So let me just subtract c one from both sides. So if I just subtract c one from both sides, I have an arbitrary, so this is going to cancel, and I have c two, sorry, let me, so this is c one, so these are going to cancel, and c two minus c one, these are both constants, arbitrary constants, we don't know what they are ... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | So it tells us when x is zero, y needs to be equal to one. So we would have one squared, which is just one, over two is equal to one half, e to the negative zero squared, well that's just going to be, e to the zero is just one, so it's going to be one half plus c, and just like that, we're able to figure out if you sub... |
Separable differential equations introduction First order differential equations Khan Academy.mp3 | Now we can multiply both sides by two, and we're going to get y squared, y squared, let me do that, so we're going to get y squared is equal to, is equal to e to the negative x squared. Now we can take the square root of both sides, and you could say, well look, y squared is equal to this, so y could be equal to the pl... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | We were in the midst of figuring out the Laplace transform of sine of at when I was running out of time. And so this is the definition of the Laplace transform of sine of at. I said that also equals y. This is going to be useful for us. It's going to be doing integration by parts twice. So I did integration by parts on... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | This is going to be useful for us. It's going to be doing integration by parts twice. So I did integration by parts once, then integration by parts twice. I said, don't worry about the boundaries of the integral right now. Let's just worry about the indefinite integral. And then after we solve for y, let's just say y i... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | I said, don't worry about the boundaries of the integral right now. Let's just worry about the indefinite integral. And then after we solve for y, let's just say y is the indefinite version of this, then we can evaluate the boundaries. And we got to this point, and we made the realization after doing two integration by... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | And we got to this point, and we made the realization after doing two integration by parts and being very careful not to hopefully make any careless mistakes, we realized, wow, this is our original y. If I put the boundaries here, that's the same thing as the Laplace transform of sine of at. That's our original y. So n... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So now, and I'll switch colors just to avoid monotony, this is equal to y. That was our original definition. So let's add a squared over sine squared y to both sides of this. So this is equal to y plus, I'm just adding this whole term to both sides of this equation, plus a squared over s squared y is equal to, so this ... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So this is equal to y plus, I'm just adding this whole term to both sides of this equation, plus a squared over s squared y is equal to, so this term is now gone, so it's equal to this stuff. And let's see if we can simplify this. So let's see if we can, let's factor out an e to the minus st. Actually, let's factor out... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So it's minus e to the minus st times sine of, well, actually, let's factor out it. Well, let me just write. 1 over s sine of at minus 1 over s squared cosine of at. I really hope I haven't made any careless mistakes. And so this, we can add the coefficients. So we get 1 plus a squared over s squared times y. But that'... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | I really hope I haven't made any careless mistakes. And so this, we can add the coefficients. So we get 1 plus a squared over s squared times y. But that's the same thing as s squared over s squared plus a squared over s squared. So it's s squared plus a squared over s squared y is equal to e minus e to the minus st ti... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | But that's the same thing as s squared over s squared plus a squared over s squared. So it's s squared plus a squared over s squared y is equal to e minus e to the minus st times this whole thing, sine of at minus 1 over s squared cosine of at. And now, this right here, since we're dealing everything with respect to dt... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So we could say a constant times the antiderivative is equal to this. This is as good a time as any to evaluate the boundaries. If this had a t here, I would have to somehow get them back on the other side, because the t's are involved in evaluating the boundaries, since we're doing our definite integral or improper in... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So let's evaluate the boundaries now. And we could have kept them along with us the whole time, right, and just factored out this term right here. But anyway, so let's evaluate this from 0 to infinity, and this should simplify things. So the right-hand side of this equation, when I evaluated at infinity, what is e to t... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So the right-hand side of this equation, when I evaluated at infinity, what is e to the minus infinity? Well, that is 0. We've established that multiple times. And now it approaches 0 from the negative side, but it's still going to be 0, or it approaches 0. And then that times, well, what's sine of infinity? Well, sine... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | And now it approaches 0 from the negative side, but it's still going to be 0, or it approaches 0. And then that times, well, what's sine of infinity? Well, sine just keeps oscillating, right, between negative 1 and plus 1, and so does cosine, right? So this is bounded. So this thing is going to overpower these. And if ... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So this is bounded. So this thing is going to overpower these. And if you're curious, you can graph it. This kind of forms an envelope around these oscillations. So the limit as this approaches infinity is going to be equal to 0. And that makes sense, right? These are bounded between 0 and negative 1, and this approach... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | This kind of forms an envelope around these oscillations. So the limit as this approaches infinity is going to be equal to 0. And that makes sense, right? These are bounded between 0 and negative 1, and this approaches 0 very quickly. So it's 0 times something bounded between 1 and negative 1. Another way to view it is... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | These are bounded between 0 and negative 1, and this approaches 0 very quickly. So it's 0 times something bounded between 1 and negative 1. Another way to view it is the largest value this could equal is 1 times whatever coefficient's on it, and this is going to 0, so it's like 0 times 1. Anyway, I don't want to focus ... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Anyway, I don't want to focus too much on that. You can play around with that if you like. Minus this whole thing evaluated at 0. So what's e to the minus 0? Well, that is e to the minus 0 is 1, right? That's e to the 0. We have a minus 1, so it becomes plus 1 times, now, sine of 0 is 0 minus 1 over s squared cosine of... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So what's e to the minus 0? Well, that is e to the minus 0 is 1, right? That's e to the 0. We have a minus 1, so it becomes plus 1 times, now, sine of 0 is 0 minus 1 over s squared cosine of 0. So what is 1 over cosine of 0? It's 1. So we have minus 1 over s squared times 1. |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | We have a minus 1, so it becomes plus 1 times, now, sine of 0 is 0 minus 1 over s squared cosine of 0. So what is 1 over cosine of 0? It's 1. So we have minus 1 over s squared times 1. So that is equal to minus 1 over s squared. And I think I made a mistake, because I shouldn't be having a negative number here. So let'... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So we have minus 1 over s squared times 1. So that is equal to minus 1 over s squared. And I think I made a mistake, because I shouldn't be having a negative number here. So let's backtrack and let's see where that mistake might be. Maybe this isn't a negative number. Let's see. Infinity, right? |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So let's backtrack and let's see where that mistake might be. Maybe this isn't a negative number. Let's see. Infinity, right? This whole thing is 0. Minus, let's see, when you put 0 here, this becomes a minus 1. Yeah, I'm getting a, let me see where my, so either this is a plus or this is a plus. |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Infinity, right? This whole thing is 0. Minus, let's see, when you put 0 here, this becomes a minus 1. Yeah, I'm getting a, let me see where my, so either this is a plus or this is a plus. Let's see where I made my mistake. e to the minus st. Oh, I see where my mistake is, right? Up here, where I factored out a minus e... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Yeah, I'm getting a, let me see where my, so either this is a plus or this is a plus. Let's see where I made my mistake. e to the minus st. Oh, I see where my mistake is, right? Up here, where I factored out a minus e to the minus st, fair enough. So that makes this 1 over s sine of at. But if I factor out a minus e to... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Up here, where I factored out a minus e to the minus st, fair enough. So that makes this 1 over s sine of at. But if I factor out a minus e to the minus st, this becomes a plus, right? There was a minus here, but I'm factoring out a minus e to the minus st. So that's a plus. This is a plus. Boy, I'm glad that was not t... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | There was a minus here, but I'm factoring out a minus e to the minus st. So that's a plus. This is a plus. Boy, I'm glad that was not too difficult to find. So then this becomes a plus, and then this becomes a plus. Thank God. It would have been sad if I wasted two videos and ended up with a careless negative number. |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Boy, I'm glad that was not too difficult to find. So then this becomes a plus, and then this becomes a plus. Thank God. It would have been sad if I wasted two videos and ended up with a careless negative number. Anyway, so now we have s squared plus a squared over s squared times y is equal to this. Multiply both sides... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | It would have been sad if I wasted two videos and ended up with a careless negative number. Anyway, so now we have s squared plus a squared over s squared times y is equal to this. Multiply both sides times s squared over s squared plus a squared, divide both sides by this, and we get y is equal to 1 over s squared. Ac... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Actually, let me make sure that that is right. It's 1 over s squared. y is equal to 1 over s squared times s squared over s squared plus a squared, and then these cancel out. And let me make sure that I haven't made another careless mistake, because I have a feeling I have. I have a feeling I have. Yep, there. I see th... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | And let me make sure that I haven't made another careless mistake, because I have a feeling I have. I have a feeling I have. Yep, there. I see the careless mistake. And it was all in this term. And I hope you don't mind my careless mistakes, but I want you to see that I'm doing these things in real time, and I'm human,... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | I see the careless mistake. And it was all in this term. And I hope you don't mind my careless mistakes, but I want you to see that I'm doing these things in real time, and I'm human, in case you haven't realized already. Anyway, so I made the same careless mistake. So I factored out e to the minus st here. So it's plu... |
Part 2 of the transform of the sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Anyway, so I made the same careless mistake. So I factored out e to the minus st here. So it's plus, but it was a over s squared. So this is an a. That's an a. And so this is an a. And so this is an a. |
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