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How would I solve this Inverse Laplace transform?

$$\mathscr{L}_s^{-1} \left\{ \frac{s}{s^2-s+\frac{17}{4}} \right\}$$

The solution is $$f(t) = (1/4 )e^{t/2} (\sin(2 t)+4 \cos(2 t))$$

I know I need to break up $F(s)$ into more common Laplace transforms, but I'm not quite sure how to begin.

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    $\begingroup$ Partial fractions, completing the square, or both. $\endgroup$
    – Chappers
    Commented Apr 28, 2015 at 2:06
  • $\begingroup$ Use the derivative property of Laplace transform to get the final result after partial fractions $\endgroup$
    – Oliver
    Commented Apr 28, 2015 at 2:09
  • $\begingroup$ Like Chappers said, complete the square in the numerator. You get $$\frac{s}{(s - \frac{1}{2})^{2} + 4}$$ which hopefully looks familiar to you. $\endgroup$ Commented Apr 28, 2015 at 2:16
  • $\begingroup$ That was it! I can't believe I didn't see it. Thank you. $\endgroup$
    – WHY
    Commented Apr 28, 2015 at 2:49
  • $\begingroup$ @Mattos, do you mean "denominator"? $\endgroup$
    – JRN
    Commented Apr 28, 2015 at 2:54

1 Answer 1

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$$\begin{align} F(s)&=\frac{s}{(s - \frac{1}{2})^{2} + 4}\\ &=\frac{\frac{1}{2}}{(s - \frac{1}{2})^{2} + 4} + \frac{s-\frac{1}{2}}{(s - \frac{1}{2})^{2}+ 4}\\ \\ f(t) &= (1/4)e^{t/2} (\sin(2 t) + 4\cos(2 t)) \end{align} $$

Credit to Chappers, Oliver, and Mattos for guiding me to the above solution

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  • $\begingroup$ ...accept yourself! $\endgroup$
    – draks ...
    Commented Jan 31, 2018 at 15:11

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