Differential Equation (Laplace transform involving a convolution)

[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.

 

[MidgetDwarf]

unit piecewise defined function, not unit function.

 

[vela]

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?

 

[MidgetDwarf]

Is there an easy guide to learning latex?

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

 

[vela]

https://www.physicsforums.com/help/latexhelp/

 

[NascentOxygen]

Is there an easy guide to learning latex?

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

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

 

[MidgetDwarf]

https://www.physicsforums.com/help/latexhelp/

Thanks,

Mods, please do not close this thread. I will uploaded my response tomorrow. I have been busy with my other course material.