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Now that we know a little bit about the convolution integral and how it applies to the Laplace transform, lets actually try to solve an actual differential equation using what we know. So I have this equation here, this initial value problem, where it says that the second derivative of y plus 2 times the first derivative of y, plus 2 times y, is equal to sine of alpha t. And they give us some initial conditions. They tell us that y of 0 is equal to 0, and that y prime of 0 is equal to 0. And thats nice and convenient that those initial conditions tend to make the problem pretty clean. But lets get to the problem. So the first thing we do is we take the Laplace transform of both sides of this equation. The Laplace transform of the second derivative of y is just s squared. This should be a bit of second nature to you by now. Its s squared times the Laplace transform of Y, which Ill just write as capital Y of s, minus s-- so we start with the same degree as the number of derivatives