Partial Fraction Decomposition with exponent in numerator

Question

$$Y(s)=\frac{1-e^\frac{-\pi*s}{2}}{s^2[(s+\frac{1}{2})^2+1]}$$

This is actually a step in an differential equations problem. I need to decompose this so I can solve the ODE.

I know how to solve the ODE. My only problem is dealing with this partial fraction - I've never encountered one that has an exponent in the numerator.

Any ideas?

Answer 1

$$Y(s)=\frac{1-e^\frac{-\pi*s}{2}}{s^2[(s+\frac{1}{2})^2+1]}=\left(1-e^\frac{-\pi*s}{2}\right)\color{blue}{\left(\frac{1}{s^2[(s+\frac{1}{2})^2+1]}\right)}$$ Off to the side you can compute the partial fraction decomposition of the blue expression (separately), return with the answer, and inverse Laplace transform the pieces from there to obtain the solution $y(t)$ of your ODE.