MHB Laplace Convolution: f(t)=-5t^2+9

[Alex2124]

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

 

[Prove It]

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) $.

 

[soloenergy]

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?