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

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?

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  • $\begingroup$ Ah! I didn't know I could factor that out before doing the decomposition. How simple. thanks man. $\endgroup$
    – 123
    Dec 2, 2014 at 3:09
  • $\begingroup$ wrong place of comment? $\endgroup$
    – BCLC
    May 22, 2016 at 4:33

1 Answer 1

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

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