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So I am working on solving the following Volterra integral equation...

$$f(t)=te^t+\int_0^t\tau f(t-\tau)d\tau$$

I take the Laplace transform of it and then solve for $F(s)$ and arrive at...

$$F(s)=\frac{s^2}{(s-1)^2(s^2-1)}$$

Now I obviously need to take the inverse Laplace transform but am not sure how to do so. Normally I would try partial fractions but I don't see how that would help here. Can anyone please explain to me what I am missing and maybe give a brief explanation of how to solve this problem?

Jake Young
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  • If partial fractions don't help here, they don't help ever. But it's more convenient to have others do the tedious calculations. –  Dec 18 '17 at 07:20

1 Answers1

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$$F(s)=\frac{s^2}{(s-1)^2(s^2-1)}=\frac{s^2}{(s-1)^3(s+1)}=\frac{A}{s-1}+\frac{B}{(s-1)^2}+\frac{C}{(s-1)^3}+\frac{D}{s+1}$$ where $A=\frac{1}{8}$, $B=\frac{3}{4}$, $C=\frac{1}{2}$, $D=-\frac{1}{8}$.

$$f(t)=Ae^t+Bte^t+C\frac{t^2}{2}e^t+De^{-t}$$

Tomi27
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