I would need a suggestion (if you have one) about the following problem: I am trying to solve a system of differential equations with non-constant coefficients. Applying the Laplace transform and solving the corresponding algebraic system I get something like this:
$$ a (s) \, F (s-s_0) + b (s) \, F (s) + c (s) \, F (s + s_0) = d (s) $$
where $s$ is the variable in the dual space of Laplace, $s_0$ is a complex number, $F$ is the transform of the function that is solution of the differential equation while $a (s)$, $b (s)$, $c (s)$, $d (s)$ are known polynomials in the dual space variable. My problem is that $F$, not known, appears to be evaluated in $s$ and in the translations $s-s_0$ and $s+s_0$.
Do you have any suggestions for solving like $F (s) = p (s) / q (s)$ where $p (s)$ and $q (s)$ will be known polynomials and then this will be inverse-Laplace-transformed? Any other ideas to solve? Any suggestion for other method to solve the problem? I have no initial information about a link among $F (s-s_0)$, $F (s)$, and $F (s+s_0)$: it should not be necessary. Is it right? Other comments/suggestions?
Thanks a lot in advance for every answer,
Kel