MHB Laplace Convolution: f(t)=-5t^2+9

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The discussion centers on the Laplace convolution of the function f(t) = -5t^2 + 9, which involves integrating f(t-u) multiplied by sin(9u). The Laplace transform of f(t) is expressed as F(s), leading to the equation F(s) = -5(2/s^3) + 9F(s)(9/(s^2 + 81)). Participants express interest in the implications of this equation, particularly how the quadratic function interacts with the integral. The conversation seeks further clarification on the nature of f(t) and its relationship to the integral component. Overall, the thread emphasizes the mathematical intricacies of the Laplace transform and convolution.
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?
 

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