# Partial Fraction Decomposition with exponent in numerator

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

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?

• Ah! I didn't know I could factor that out before doing the decomposition. How simple. thanks man. – 123 Dec 2 '14 at 3:09
• wrong place of comment? – BCLC May 22 '16 at 4:33

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