MHB Laplace Convolution: f(t)=-5t^2+9

Alex2124
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f(t)=-5t^2+9\int_{0}^{t} \,f(t-u)sin(9u)du
 

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Alex2124 said:
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) $.
 


I find this equation to be quite interesting. It looks like a combination of a quadratic function and an integral. I'm curious to know what the function f(t) represents and how it relates to the integral in the equation. Can you provide any more context or information about this equation?
 
I posted this question on math-stackexchange but apparently I asked something stupid and I was downvoted. I still don't have an answer to my question so I hope someone in here can help me or at least explain me why I am asking something stupid. I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem: Let ##f:D\to\mathbb{C}## be an anlytic function over a simply connected region ##D##. If ##a## and ##z## are part of...

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