Saturday 25 January 2014

Tough inverse Laplace transform



I know what the solution is to this inverse Laplace transform, I just have NO idea how to get there.



$$\mathcal{L}^{-1}\left(\frac{16s}{\left(s^2+4\right)^2}\right)$$



Basically, my question is what modification do I have to do to the equation above?


Answer




Using a table, note the form:



$$f(t) = t\sin(at)$$



$$F(s) = \frac{2as}{(s^2+a^2)^2}$$



Using this:
$$f(t) = \mathcal{L}^{-1}\left(\frac{16s}{\left(s^2+4\right)^2}\right) = \mathcal{L}^{-1}\left(\frac{2*2*s}{\left(s^2+2^2\right)^2}*4\right) =4t\sin(2t)$$


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