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I am trying to solve the damped harmonic oscillator:

[itex] \frac{d^2y}{dt^2}+\frac{b}{m} \frac{dy}{dt}+\frac{k}{m}y=0 [/itex]

and I thought using the Laplace transform might do the trick. Anyway so I did the LT (and inserted the initial conditions that at t=0 y=A, and dy/dt=0) and obtained:

[itex] Y(p)=\frac{A(p+\frac{b}{m})}{p^2+\frac{b}{m}p+\frac{k}{m} } [/itex]

I can't seem to get this into a simple form so I can use the lookup tables to do the inverse transform. So I guess I have to somehow use the Bromwhich integral. Obviously the two poles are [itex]p=-\frac{b}{2m} \pm sqrt{ \frac{b^2}{4m^2}-\frac{k}{m} [/itex] and I recognise this from the actual solution so I think I'm on the right track.

I just don't know how to get the residues and complete the inverse

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# Damped HO and Laplace transform method

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