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L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Remember, the Laplace transform is just a definition, it's just a tool that has turned out to be extremely useful. And we'll do more on that intuition later on. But anyways, the integral from 0 to infinity of e to the minus st times whatever we're taking the Laplace transform of times sine of at dt. And now we have to ... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | And now we have to go back and find our integration by parts neuron, and mine always disappears. So we have to reprove integration by parts. I don't recommend you do this all the time. If you have to do this on an exam, you might want to memorize it before the exam. But always remember, integration by parts is just the... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | If you have to do this on an exam, you might want to memorize it before the exam. But always remember, integration by parts is just the product rule in reverse. So I'll just do that in this corner. So the product rule tells us we have two functions, u times v, and if I were to take the derivative of u times v, let's sa... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So the product rule tells us we have two functions, u times v, and if I were to take the derivative of u times v, let's say that they're functions of t. These are both functions of t. I could have written u of x times v of x. That equals the derivative of the first times the second function plus the first function time... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Of u prime v with respect to dt, but I'm just doing a little bit of shorthand now. Plus the integral of uv prime. I'm just trying to help myself remember this thing. And let's take this and subtract it from both sides. So we have this integral, the integral of u prime v is going to be equal to this, uv minus the integr... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | And let's take this and subtract it from both sides. So we have this integral, the integral of u prime v is going to be equal to this, uv minus the integral of uv prime. And of course, this is a function of t. There's a dt here and all of that. But I just have to do this in the corner of my page a lot, because I always... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | But I just have to do this in the corner of my page a lot, because I always forget this. And with the primes and the integrals and all that, I always forget it. One way, if you did want to memorize it, you said OK, integration by parts says if I take the integral of the derivative of one thing and then just a regular f... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Here, when you take the subtraction, you're taking the one that had a derivative now doesn't, and the one that didn't have a derivative now does. But anyway, let's apply that to our problem at hand, to this one. Well, let's make, we could go either way about it. Let's make u prime is equal to, so let's say u prime, we'... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Let's make u prime is equal to, so let's say u prime, we'll do our definition. u prime is equal to e to the minus st, in which case u would be the antiderivative of that, which is equal to minus 1 over s e to the minus st. All right? And actually, just so, this is going to be an integration by parts twice problem. So I... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So I'm just actually going to define the Laplace transform as y. That'll come in useful later on. And I think I actually did a very similar example to this when we did integration by parts. But anyway, back to the integration by parts. So that's u, and let me do v in a different color. So in v, if this is u prime, then... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | But anyway, back to the integration by parts. So that's u, and let me do v in a different color. So in v, if this is u prime, then this is v. So v is equal to sine of at. And then what is v prime? Well, that's just a cosine of at, the chain rule. And now we're ready to do our integration. So the Laplace transform, and ... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | And then what is v prime? Well, that's just a cosine of at, the chain rule. And now we're ready to do our integration. So the Laplace transform, and I'll just say that's y. y is equal to, right? y is what we're trying to solve for, the Laplace transform of sine of at. That is equal to u prime v. We had to find u prime ... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So the Laplace transform, and I'll just say that's y. y is equal to, right? y is what we're trying to solve for, the Laplace transform of sine of at. That is equal to u prime v. We had to find u prime in v, right? That's equal to that. The integral of u prime times v. That equals uv. So uv, so that's minus 1 over se to... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | That's equal to that. The integral of u prime times v. That equals uv. So uv, so that's minus 1 over se to the minus st times v sine of at minus the integral. And when you do the integration by parts, this could be an indefinite integral, an improper integral, a definite integral, whatever. But the boundaries stay. So ... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | And when you do the integration by parts, this could be an indefinite integral, an improper integral, a definite integral, whatever. But the boundaries stay. So we could still say from 0 to infinity. Of uv prime. So u is minus 1 over se to the minus st times v prime. Times a cosine of at. Fair enough. |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Of uv prime. So u is minus 1 over se to the minus st times v prime. Times a cosine of at. Fair enough. dt. Well now we have another hairy integral we need to solve. So this might involve another integration by parts. |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Fair enough. dt. Well now we have another hairy integral we need to solve. So this might involve another integration by parts. And it does. So here, let's see if we can simplify it. Let's take the constants out first. |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So this might involve another integration by parts. And it does. So here, let's see if we can simplify it. Let's take the constants out first. Let me just rewrite this. So we get y is equal to minus e to the minus st over s sine of at. So you have a minus minus plus a over s. Plus a over s. Right? |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Let's take the constants out first. Let me just rewrite this. So we get y is equal to minus e to the minus st over s sine of at. So you have a minus minus plus a over s. Plus a over s. Right? a divided by s. And these two negative signs cancel out. Times the integral from 0 to infinity e to the minus st cosine of at dt... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | So you have a minus minus plus a over s. Plus a over s. Right? a divided by s. And these two negative signs cancel out. Times the integral from 0 to infinity e to the minus st cosine of at dt. Let's do another integration by parts. And I'll do this in a purple color. Just so you know, this is our second integration by ... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Let's do another integration by parts. And I'll do this in a purple color. Just so you know, this is our second integration by parts. Let's define, once again, u prime is equal to e to the minus st. So this is u prime. Then u is equal to minus 1 over se to the minus st. We'll make v equal to cosine of at. The hardest p... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Let's define, once again, u prime is equal to e to the minus st. So this is u prime. Then u is equal to minus 1 over se to the minus st. We'll make v equal to cosine of at. The hardest part about this is just not making careless mistakes. And then v prime, I just want it to be in the same row, is equal to minus a sine ... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | The hardest part about this is just not making careless mistakes. And then v prime, I just want it to be in the same row, is equal to minus a sine of at. Right? Chain rule, derivative of cosine is minus sine. So let's substitute that back in. And we get, this is going to get hairy. Actually, it already is hairy. |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Chain rule, derivative of cosine is minus sine. So let's substitute that back in. And we get, this is going to get hairy. Actually, it already is hairy. y is equal to minus e to the minus st over s sine of at plus a over s times integration by parts. uv, so that's minus 1 over se to the minus st times v times cosine at... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Actually, it already is hairy. y is equal to minus e to the minus st over s sine of at plus a over s times integration by parts. uv, so that's minus 1 over se to the minus st times v times cosine at minus the integral from 0 to infinity. This problem's making me hungry. It's taking so much glucose from my bloodstream. ... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | This problem's making me hungry. It's taking so much glucose from my bloodstream. I'm focusing so much not to make careless mistakes. Anyway, integral from 0 to infinity. And now we have uv prime. So u is minus 1 over s e to the minus st. That's u. And then v prime times minus a. |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Anyway, integral from 0 to infinity. And now we have uv prime. So u is minus 1 over s e to the minus st. That's u. And then v prime times minus a. So let's make that minus cancel out with this one so that becomes a plus. a sine of at dt. I'm starting to see the light at the end of the tunnel. |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | And then v prime times minus a. So let's make that minus cancel out with this one so that becomes a plus. a sine of at dt. I'm starting to see the light at the end of the tunnel. So then, let's simplify this thing. And of course, we're going to have to evaluate this whole thing, right? From infinity. |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | I'm starting to see the light at the end of the tunnel. So then, let's simplify this thing. And of course, we're going to have to evaluate this whole thing, right? From infinity. Actually, we're going to have to evaluate everything. Let's just focus on the indefinite integral for now. We're going to have to take this w... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | From infinity. Actually, we're going to have to evaluate everything. Let's just focus on the indefinite integral for now. We're going to have to take this whole thing and evaluate. Let's just say that y is the antiderivative and then evaluate it from infinity to 0, from 0 to infinity. So this is equal to y is equal to ... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | We're going to have to take this whole thing and evaluate. Let's just say that y is the antiderivative and then evaluate it from infinity to 0, from 0 to infinity. So this is equal to y is equal to minus e to the minus st over s sine of at. Now let's distribute this. Minus a over s squared e to the minus st cosine of a... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Now let's distribute this. Minus a over s squared e to the minus st cosine of at. Right? OK, now I want to make sure I don't make a careless mistake. Now let's multiply this times this and take all the constants out. So we have an a and an s. a over s. There's a minus sign. We have a plus a to the s. So we'll have a mi... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | OK, now I want to make sure I don't make a careless mistake. Now let's multiply this times this and take all the constants out. So we have an a and an s. a over s. There's a minus sign. We have a plus a to the s. So we'll have a minus a squared over s squared times the integral from 0. Well, I said I'm just worrying ab... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | We have a plus a to the s. So we'll have a minus a squared over s squared times the integral from 0. Well, I said I'm just worrying about the indefinite integral right now and we'll evaluate the boundaries later. e to the minus st sine of at dt. Now this is the part, and we've done this before, it's a little bit of a t... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Now this is the part, and we've done this before, it's a little bit of a trick with integration by parts. But this expression, notice, is the same thing as our original y. Right? This is our original y and we're assuming we're doing the indefinite integral and we'll evaluate the boundaries later. Although we could have... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | So the first thing I want to introduce is just kind of a quick way of doing something. And that is, if I had the Laplace transform of the second derivative of y. Well, we've proved several videos ago that if I wanted to take the Laplace transform of the first derivative of y, that is equal to s times the Laplace transf... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | And we used this property in the last couple of videos to actually figure out the Laplace transform of the second derivative. Because if you say this is y prime, this is the antiderivative of it, then you can just pattern match. You could say, well, the Laplace transform of y prime that's just equal to s times the Lapl... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | This is the derivative of this, just like this is the derivative of this. I'll draw a line here just so you don't get confused. So Laplace transform of y prime prime is this thing. And now we can use this, which we proved several videos ago, to resubstitute it and get in terms of the Laplace transform of y. So we can e... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | And now we can use this, which we proved several videos ago, to resubstitute it and get in terms of the Laplace transform of y. So we can expand this part. The Laplace transform of the derivative of y, that's just equal to s times the Laplace transform of y minus y of 0. And then we have the outside. We have s minus y ... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | And then we have the outside. We have s minus y prime of 0. And then when you expand it all out, and we've done this before, you get s squared times the Laplace transform of y minus s times y of 0 minus y prime of 0. Now there's something interesting to note here. And if you learn this, it'll make it a lot faster. You ... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | Now there's something interesting to note here. And if you learn this, it'll make it a lot faster. You won't have to go through all this and risk making careless mistakes when you have scarce time and paper on your tests. Just notice that when you take the Laplace transform of the second derivative, what do we end up? ... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | Just notice that when you take the Laplace transform of the second derivative, what do we end up? We end up with s squared. This was the second derivative, so we end up with s squared times the Laplace transform of y minus s times y of 0 minus 1 times y prime of 0. So every term, we started with s squared. And then eve... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | So every term, we started with s squared. And then every term, we lowered the degree of s1. And then everything except the first term is a negative sign. And then we started with the Laplace transform of y. And then you can almost view the Laplace transform as a kind of integral. So we kind of take the derivative. So t... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | And then we started with the Laplace transform of y. And then you can almost view the Laplace transform as a kind of integral. So we kind of take the derivative. So then you get y. And then you take the derivative again. You get y prime. And of course, every other term is negative. |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | So then you get y. And then you take the derivative again. You get y prime. And of course, every other term is negative. And these aren't the actual functions. These are those functions evaluated at 0. But that's a good way to help you hopefully remember how to do these. |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | And of course, every other term is negative. And these aren't the actual functions. These are those functions evaluated at 0. But that's a good way to help you hopefully remember how to do these. And once you get the hang of it, you can take the Laplace transform of any arbitrary function very, very quickly. Or any arb... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | But that's a good way to help you hopefully remember how to do these. And once you get the hang of it, you can take the Laplace transform of any arbitrary function very, very quickly. Or any arbitrary derivative. So let's say we wanted to take the Laplace transform of, I don't know, this should hit the point home, the ... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | So let's say we wanted to take the Laplace transform of, I don't know, this should hit the point home, the fourth derivative of y. That 4 in parentheses means the fourth derivative. I could have drawn 4 prime marks, but either way. So what is this equal to? If we use this technique and substitute it, we're bound to mak... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | So what is this equal to? If we use this technique and substitute it, we're bound to make some form of careless mistake or other. And it would take us forever. And it would waste a lot of paper. But now we see the pattern. And so we can just say, well, the Laplace transform of this, in terms of the Laplace transform of... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | And it would waste a lot of paper. But now we see the pattern. And so we can just say, well, the Laplace transform of this, in terms of the Laplace transform of y, that's what we want to get to, is going to be s to the fourth times the Laplace transform of y. Now every other term's going to have a minus in front of it.... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | Now every other term's going to have a minus in front of it. Minus, lower the degree on the s, minus s to the third. And then you could kind of say, let's take some form of derivative so that you get y of 0. It's not a real derivative. The Laplace transform really isn't the antiderivative of y of 0. But anyway, I think... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | It's not a real derivative. The Laplace transform really isn't the antiderivative of y of 0. But anyway, I think you get the idea. And then we lower the degree on s again. Minus s squared. Take the derivative. And of course, these aren't functions. |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | And then we lower the degree on s again. Minus s squared. Take the derivative. And of course, these aren't functions. But we're evaluating the derivative of that function now at 0. So y prime of 0. Minus, now we lower the degree one more, minus s times, this is an s, times y prime prime of 0. |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | And of course, these aren't functions. But we're evaluating the derivative of that function now at 0. So y prime of 0. Minus, now we lower the degree one more, minus s times, this is an s, times y prime prime of 0. We have one more term. Lower the degree on the s one more time. Then you get s to the 0, which is just 1. |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | Minus, now we lower the degree one more, minus s times, this is an s, times y prime prime of 0. We have one more term. Lower the degree on the s one more time. Then you get s to the 0, which is just 1. So minus 1 is a coefficient. And then you have y, the third derivative of y. Let me scroll over a little bit. |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | Then you get s to the 0, which is just 1. So minus 1 is a coefficient. And then you have y, the third derivative of y. Let me scroll over a little bit. The third derivative of y evaluated at 0. So I think you see the pattern now. And this is a much faster way of evaluating the Laplace transform of an arbitrary derivati... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | Let me scroll over a little bit. The third derivative of y evaluated at 0. So I think you see the pattern now. And this is a much faster way of evaluating the Laplace transform of an arbitrary derivative of y, as opposed to keep going through that pattern over and over again. Another thing I want to introduce you to is... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | And this is a much faster way of evaluating the Laplace transform of an arbitrary derivative of y, as opposed to keep going through that pattern over and over again. Another thing I want to introduce you to is just a notational savings. And it's just something that you'll see. So you might as well get used to it. And i... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | So you might as well get used to it. And it actually saves time over keep writing this curly l in this bracket. If I, the Laplace transform of y of t, I can write as, and people tend to write it as, well, it's going to be a function of s. And what they use is they use a capital Y to denote the function of s. And that, ... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | Because normally, when we're doing antiderivatives, you just take, you know, when we take, we often, when you learn the fundamental theorem of calculus, learn that the integral of f with respect to dx, you know, from 0 to x, is equal to capital F of x. So it's kind of borrowing that notation, because this function of s... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | When you see capital Y of s, that's the same thing as the Laplace transform of y of t. And you might also see it this way. The Laplace transform of f of t is equal to capital F of s. And the clue that tells you that this isn't just a normal antiderivative is the fact that they're using that s as the independent variabl... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | Anyway, I'm trying to think whether I have time to teach you more fascinating concepts of Laplace transform. Well, sure, I think we do. So my next question for you, and now we'll teach you a couple more properties, and this will be helpful in taking Laplace transforms. What is the Laplace transform of e to the at times... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | What is the Laplace transform of e to the at times f of t? Fascinating. Well, let's just go back to our definition of the Laplace transform. It is the integral from 0 to infinity of e to the minus st times whatever we have between the curly brackets. So with the curly brackets, we have e to the at f of t dt. And now we... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | It is the integral from 0 to infinity of e to the minus st times whatever we have between the curly brackets. So with the curly brackets, we have e to the at f of t dt. And now we can add these exponents, right? We have a similar base. So this is equal to what? This is equal to the integral from 0 to infinity. And let'... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | We have a similar base. So this is equal to what? This is equal to the integral from 0 to infinity. And let's see, I want to write it as, I could write it minus s plus a, but I'm going to write it as minus s minus at. And you could expand this out, right? It becomes minus s plus a, which is exactly what we have here, t... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | And let's see, I want to write it as, I could write it minus s plus a, but I'm going to write it as minus s minus at. And you could expand this out, right? It becomes minus s plus a, which is exactly what we have here, times f of t dt. Now let me show you something. If I were to just take the Laplace transform of f of ... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | Now let me show you something. If I were to just take the Laplace transform of f of t, that is equal to some function of s. Whatever we essentially have right here for s, it becomes some function of that. So this is interesting. This is some function of s. Here all we did, to go from, well actually, let me rewrite this... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | This is some function of s. Here all we did, to go from, well actually, let me rewrite this, so Laplace, which is equal to 0 to infinity e to the minus s t f of t dt. The Laplace transform of just f of t is equal to this, which is some function of s, right? Well the Laplace transform of e to the at times f of t, it equ... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | And what's the difference between this and this? What's the difference between the two? Well it's not much. Here, wherever I have an s, I have an s minus a here, right? So if this is a function of s, what's this going to be? It's going to be that same function. Whatever the Laplace transform of f was, it's going to be ... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | Here, wherever I have an s, I have an s minus a here, right? So if this is a function of s, what's this going to be? It's going to be that same function. Whatever the Laplace transform of f was, it's going to be that same function, but instead of s, it's going to be a function of s minus a. And once again, how did I ge... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | Whatever the Laplace transform of f was, it's going to be that same function, but instead of s, it's going to be a function of s minus a. And once again, how did I get that? Well I said the Laplace transform of f is a function of s, and it's equal to this. Well if I just replaced an s with an s minus a, I get this, whi... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | Well if I just replaced an s with an s minus a, I get this, which is a function of s minus a, which was the Laplace transform of e to the at times f of t. Maybe that's a little confusing. Let me show you an example. So if I, let's just take the Laplace transform of cosine of 2t, we've shown is equal to, well I'll write... |
Shifting transform by multiplying function by exponential Differential Equations Khan Academy.mp3 | We've shown that already. And so the Laplace transform of e to the, I don't know, 3t times cosine of 2t is going to be equal to the same function, but instead of s, it's going to be a function of s minus a, so s minus 3, which is equal to s minus 3 over s minus 3 squared plus 4. Notice, when you just multiply something... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | But in order to just kind of make sure we don't get confused, I think it might be useful to review a little bit of everything that we've learned so far. So in the last video, we saw that the Laplace transform, well, let me just write something a little, the Laplace transform of f of t, let me just get some notation dow... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | And that one told us that the Laplace transform, the Laplace transform of e to the a t times f of t, that this is equal to, instead of, and I wanna make this distinction very clear, here we shifted the f of t, and we got just a kind of a regular f of s. In this situation, when we multiply it times the e to the positive... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | So we're gonna do a couple of examples that we're gonna have to figure out which one of these two apply. And let's write all the other stuff that we learned as well. The very first thing we learned was that the Laplace transform of one was equal to one over s. We know that's a pretty straightforward one, easy to prove ... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | And more generally, we learned that the Laplace transform of t to the n, where n is a positive integer, n is a positive integer, it equaled n factorial over n factorial over s to the n plus one. And then we had our trig functions that we've gone over. Let me do this in a different color. Let me do it, I'll do it right ... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | Let me do it, I'll do it right here. The Laplace transform of sine of a t is equal to a over s squared plus a squared. And the Laplace transform of the cosine of a t is equal to s over s squared plus a squared. And you'll be amazed by how far we can go with just what I've written here. In future videos, we're gonna bro... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | And you'll be amazed by how far we can go with just what I've written here. In future videos, we're gonna broaden our toolkit even further. But just these right here, you can already do a whole set of Laplace transforms and inverse Laplace transforms. So let's try to do a few. So let's say I were to give you the Laplac... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | So let's try to do a few. So let's say I were to give you the Laplace transform. And this is just the hard part. I think you know how to solve a differential equation if you know how to take the Laplace transforms and go back and forth. The hard part is just recognizing or inverting your Laplace transform. So let's say... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | I think you know how to solve a differential equation if you know how to take the Laplace transforms and go back and forth. The hard part is just recognizing or inverting your Laplace transform. So let's say we had the Laplace transform of some function, f of s. Let's say it's three factorial over s minus two to the fo... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | Now, your pattern matching or your pattern recognition part of your brain should immediately say, look, I have a Laplace transform of something that has a factorial in it and it's over an exponent. This must be something related to this thing right here. If I just had the Laplace transform, let me write that down, the ... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | So if you write the Laplace transform of t to the three, this rule that we showed right here, this means that it would be equal to three factorial over s to the fourth. Now, this thing isn't exactly this thing. They're not quite the same thing. And just so, you know, I'm doing this to instruct you, but I find these whe... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | And just so, you know, I'm doing this to instruct you, but I find these when I'm actually doing them on an exam. I remember when I did them when I first learned this, I would actually go through this step because I was, you know, you definitely don't wanna make a careless mistake and you definitely wanna kinda make sur... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | And this one here. Well, we've shifted our s, right? If we call this expression right here f of s, we call this expression right here f of s, then what's this expression? This expression right here is f of s minus two, right? That expression is f of s minus two. So what are we dealing with here? So you see here you hav... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | This expression right here is f of s minus two, right? That expression is f of s minus two. So what are we dealing with here? So you see here you have a shifted f of s, right? So in this case, a would be equal to two. So this is the Laplace transform of e to the at times our f of t. So this is the Laplace transform. So... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | So you see here you have a shifted f of s, right? So in this case, a would be equal to two. So this is the Laplace transform of e to the at times our f of t. So this is the Laplace transform. So let me write this down. This is the Laplace transform of e to the, and what's a? a is what we shifted by, right? It's what we... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | So let me write this down. This is the Laplace transform of e to the, and what's a? a is what we shifted by, right? It's what we shifted by, minus a, so you have a positive there. So e to the two t times the actual function. If this was just an f of s, what would f of t be? Well, we figured that out. |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | It's what we shifted by, minus a, so you have a positive there. So e to the two t times the actual function. If this was just an f of s, what would f of t be? Well, we figured that out. It was t to the three, t to the third power. So this, the Laplace transform of this is equal to that, or we could write that the inver... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | Well, we figured that out. It was t to the three, t to the third power. So this, the Laplace transform of this is equal to that, or we could write that the inverse Laplace transform of three factorial over s minus two to the fourth minus two to the fourth is equal to e to the two t times t to the third. Now, if that se... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | Now, if that seemed confusing to you, you can kind of go forward. You can kind of, let's apply this. Let's go the other direction, and maybe this will make it a little bit clearer for you. So let's go from this direction. If I had to take the Laplace transform of this thing, I'd say, okay, well, the Laplace transform o... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | So let's go from this direction. If I had to take the Laplace transform of this thing, I'd say, okay, well, the Laplace transform of t to the third is easy. The Laplace transform of t to the third, I'll do it here, I think, yeah, the tool isn't working right there properly. Let me scroll up a little bit. So I could wri... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | Let me scroll up a little bit. So I could write it right here. So if I wanted to figure out the Laplace transform of e to the two t times t to the third, I'll say, well, you know, this e to the two t, I remember that it shifts something. So if I know that the Laplace transform of t to the third, this is an easy one, it... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | So if I know that the Laplace transform of t to the third, this is an easy one, it's equal to three factorial over s to the fourth, that's three plus one. Then the Laplace transform of e to the two t times t to the third is going to be this shifted. This is equal to f of s. Then this is going to be f of s minus two. F ... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | F of s minus two. So what's f of s minus two? It's going to be equal to three factorial over s minus two to the fourth. I think you're already getting an appreciation that the hardest thing about these Laplace transform problems are really kind of all of these shifts and kind of recognizing the patterns and recognizing... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | I think you're already getting an appreciation that the hardest thing about these Laplace transform problems are really kind of all of these shifts and kind of recognizing the patterns and recognizing what's your a and what's your c and being very careful about it so you don't make a careless mistake. I think doing a l... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | So let's try this one right here. This looks like a little bit more complicated. They give us that the Laplace transform of some function is equal to two times s minus one times e to the minus two s, all of that over s squared minus two s plus two. Now this looks very daunting. How do you do this? I have an e here, I h... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | Now this looks very daunting. How do you do this? I have an e here, I have something shifted here, I have this polynomial in the denominator here. What can I do with this? So the first thing when I look at these polynomials in the denominator, I say can I factor it somehow? And can I factor it out fairly simply? And ac... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | What can I do with this? So the first thing when I look at these polynomials in the denominator, I say can I factor it somehow? And can I factor it out fairly simply? And actually in the exams that you'll find in differential equation class, they'll never give you something that's factorable into these weird numbers. I... |
Inverse Laplace examples Laplace transform Differential Equations Khan Academy.mp3 | And actually in the exams that you'll find in differential equation class, they'll never give you something that's factorable into these weird numbers. It tends to be integers. When you say okay, what two numbers, they have to be positive. When you give their product, you get two. And then when you add them, you get ne... |
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