Question

Does anyone know how to find the Inverse Laplace Transform of 6/(s^3+3s)

Answer 1

It is combination of simple Laplace transformation:\[\frac{6}{s^3+3s}=\frac{6}{s(s^2+3)}\]
Use Method of undetermined coefficients:
\[\frac{6}{s(s^2+3)} \rightarrow \frac{2}{s}-\frac{2s}{s^2+3}\]
Use well know transformation for this functions:
\[\frac{2}{s} \rightarrow 2\cdot H(t)\], H(t) - Heaviside step function
\[\frac{2s}{s^2+3}\rightarrow 2\cdot \cos(\sqrt{3}t)\]

\[\frac{6}{s^3+3s} \rightarrow 2\cdot H(t) +2\cdot \cos(\sqrt{3}t)\]

Answer 2

we know that L[sin 3^(1/2) t]= (3^(1/2))/(s^2+3) and L[integral f(k) dk (in limits 0 to t)]=L[f(t)]/s. Here we can consider f(t)=sin 3^(1/2) t,then (3^(1/2))/s(s^2 +3) =integral f(k) dk in limits 0 to t .the integral value is 1-(cos3^(1/2) t)/(3^(1/2)).
therefore, L[6/((s^3)+3s)] = 2(1-(cos3^(1/2) t)/(3^(1/2)).