Laplace Transform solution help

[Melawrghk]

Homework Statement

Find the solution of the given initial value problem:
y''+4y=u_pi(t)-u_3pi(t) y(0)=7, y'(0)=5

The Attempt at a Solution

So I found the L{} of the above equation:
s²Y-s*f(0)-f'(0)+4Y = (e^-pi*s)/s-(e^-3pi*s)/s

Combining and substituting the numbers I get:
Y=\frac{e^{-pi*s}-e^{-3pi*s}}{s(s^{2}+4)}+\frac{6s+3}{s^2+4}

I know how to do the second term's inverse Laplace, but not the first. Here is what I tried:
I can see that I can't get rid of the exponentials in any way other than using the step function again. And the other denominator factor (s^2+4) can be potentially used to get sine. So that:
Y=\frac{1}{2}(\frac{e^{-pi*s}-e^{-3pi*s}}{s})\frac{2}{s^{2}+4}

This is where I don't know what to do. I can't separate them and I don't know of a way to do Laplace inverse of a product.

Any help would be really appreciated. Thanks

 

[Vid]

partial fractions

Also, the exponentials should be powers of s not t.

 

[Melawrghk]

How do I do partial fractions with exponentials? Do I use like Ae^(-pi*s) instead of the usual A?

Edit: I fixed the powers

 

[Vid]

Nevermind partial fractions won't help here. You need to have proven a theorem about the laplace transform of the step function times another function.

 

[Melawrghk]

Nah, I got it, you do need partial fractions. Thanks.