# Differential Equation (Laplace transform involving a convolution)

1. Nov 26, 2015

### MidgetDwarf

So I have to solve an initial value problem involving the Laplace Transformation method. I have all the terms in Y(t) besides one term, I cannot figure how to change it from frequency domain back into time domain.

Not sure how to type in Latex, so i uploaded a picture, using the whiteboard feature on this webpage.

When I try to use the definition of the convolution, where F*G, I'm not sure what to do with unit function and
cos(4t-4pie), in other words how to change them into tao.

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2. Nov 27, 2015

### MidgetDwarf

unit piecewise defined function, not unit function.

3. Nov 27, 2015

### vela

Staff Emeritus
You should spend a few minutes and learn how to use LaTeX.

You have $Y(s) = e^{-\pi s} \frac{s}{(s^2+16)^2}$. If you want to split it up and use convolution, I'd rewrite it as
$$Y(s) = \frac 14 e^{-\pi s} \left(\frac{s}{s^2+16}\right)\left(\frac{4}{s^2+16}\right).$$ You can deal with the delay at the end. So now you have $f(t) = \cos 4t$ and $g(t) = \sin 4t$. What's the definition of convolution?

4. Nov 27, 2015

### MidgetDwarf

Is there an easy guide to learning latex?

So I can post the response and make it easier for you guys.

5. Nov 27, 2015

### vela

Staff Emeritus
6. Nov 27, 2015

### Staff: Mentor

If you click on "Reply" you'll be able to see the Latex instructions that others have used to produce their post.

7. Nov 27, 2015