# Solving an lineare integro Differential equation

My integro differential equation reads:

$\dot{f}(t)=-\int_0^t\mathrm{d}s\text{ }g(t-s)f(s)$

with $g(t)=\frac{\text{exp}(i\gamma t)}{(1+it)^2}$ with $t\ge 0$, $\gamma>0$.

I tried to solve it with the Laplace-tranformation approach, which failed. The Laplace-transformation of $g$ exists, but the inverse calculation is way to complicated to get anaytically.

Is there any other apporach one could choose? The convolution kernel $g$ looks a lot like something related to the exponentialintegral (or a generalization of it). The numerical solution as well as some perturbation-like solution of the equation shows that at least it has to do something with it. I didn't find any helpfull formulas online though. There are no analytical papers so far, that deal with this kind of integro-differential equation other than the Laplace-appraoch.

Thanks for the answer already!

Martin

• I already tried the Laplace Transform Technique. It's cumbersome because we need to find roots of an expression which involve the Exponential Integral. – Felix Marin Mar 1 '18 at 5:19
• Exactly...i thought maybe there is some solution avoiding the Laplace-transformation in general. But thanks nonetheless – Martin Mar 1 '18 at 10:14