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And actually, you end up having a characteristic equation. Well, we could use this once again, so let's do that.And the initial conditions are y of 0 is equal to 2, and y prime of 0 is equal to 3. So this over here-- I'll do it in magenta-- this is equal to s times what? Well that's s times the Laplace Transform of y, minus y of 0, right?However, if we combine the two terms up we will only be doing partial fractions once.
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So, in order to find the solution all that we need to do is to take the inverse transform.
Before doing that let’s notice that in its present form we will have to do partial fractions twice.
And I've gotten a bunch of letters on the Laplace Transform. It's hard to really have an intuition of the Laplace Transform in the differential equations context, other than it being a very useful tool that converts differential or integral problems into algebra problems.
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But I'll give you a hint, and if you want a path to learn it in, you should learn about Fourier series and Fourier Transforms, which are very similar to Laplace Transforms. And that's good, because I didn't have space to do another curly L. So the Laplace Transform of y prime prime, if we apply that, that's equal to s times the Laplace Transform of-- well if we go from y prime to y, you're just taking the anti-derivative, so if you're taking the anti-derivative of y, of the second derivative, we just end up with the first derivative-- minus the first derivative at 0.We are trying to find the solution, \(y(t)\), to an IVP.What we’ve managed to find at this point is not the solution, but its Laplace transform.\[\mathcal\left\ - 10\mathcal\left\ 9\mathcal\left\ = \mathcal\left\] Using the appropriate formulas from our table of Laplace transforms gives us the following.\[Y\left( s \right) - sy\left( 0 \right) - y'\left( 0 \right) - 10\left( \right) 9Y\left( s \right) = \frac\] Plug in the initial conditions and collect all the terms that have a \(Y(s)\) in them.And that'll actually build up the intuition on what the frequency domain is all about. So let's say the differential equation is y prime prime, plus 5, times the first derivative, plus 6y, is equal to 0. So what are the Laplace Transforms of these things? Notice, we're already using our initial conditions. And then we end up with plus 5, times-- I'll write it every time-- so plus 5 times the Laplace Transform of y prime, plus 6 times the Laplace Transform of y. So just to be clear, all I did is I expanded this into this using this.Well anyway, let's actually use the Laplace Transform to solve a differential equation. And you know how to solve this one, but I just want to show you, with a fairly straightforward differential equation, that you could solve it with the Laplace Transform. So the Laplace Transform of 0 would be be the integral from 0 to infinity, of 0 times e to the minus stdt. Well this is where we break out one of the useful properties that we learned. I think that's going to need as much real estate as possible. So we learned that the Laplace Transform-- I'll do it here. The Laplace Transform of f prime, or we could even say y prime, is equal to s times the Laplace Transform of y, minus y of 0. So how can we rewrite the Laplace Transform of y prime?While Laplace transforms are particularly useful for nonhomogeneous differential equations which have Heaviside functions in the forcing function we’ll start off with a couple of fairly simple problems to illustrate how the process works.The first step in using Laplace transforms to solve an IVP is to take the transform of every term in the differential equation.Now we're just taking Laplace Transforms, and let's see where this gets us. So I get the Laplace Transform of y-- and that's good because it's a pain to keep writing it over and over-- times s squared plus 5s plus 6. Because the characteristic equation to get that, we substituted e to the rt, and the Laplace Transform involves very similar function. What I'm going to do is I'm going to solve this.And actually I just want to make clear, because I know it's very confusing, so I rewrote this part as this. I'm going to say the Laplace Transform of y is equal to something. We haven't solved for y yet, but we know that the Laplace Transform of y is equal to this.