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Laplace Convolution

Alex2124

New member
Mar 19, 2020
1
f(t)=-5t^2+9\int_{0}^{t} \,f(t-u)sin(9u)du
 

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Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
f(t)=-5t^2+9\int_{0}^{t} \,f(t-u)sin(9u)du
$\displaystyle \mathcal{L} \left\{ f\left( t \right) \right\} = F\left( s \right) $, so

$\displaystyle \begin{align*} \mathcal{L} \left\{ f\left( t \right) \right\} &= \mathcal{L}\left\{ -5\,t^2 \right\} + 9\,\mathcal{L}\left\{ \int_0^t{ f\left( t - u \right) \,\sin{\left( 9\,u \right) } \,\mathrm{d}u } \right\} \\
F\left( s \right) &= -5 \left( \frac{2}{s^3} \right) + 9 \,F\left( s \right) \left( \frac{9}{s^2 + 81} \right) \end{align*}$

Now solve for $F\left( s \right) $.