There are various techniques to find the fundamental solutions for a given linear ordinary differential equation (ode). I am interested in reverse engineering; to find a differential equation from a set of fundamental solution. However, it is easy for constant coefficients using corresponding characteristic equation but how about non constant coefficients.
For example, if we have following third order differential equation: $y'''+p(x)y''+q(x)y'+r(x)y=0$ and $\left \{1,exp(x^2), log(x) \right \}$ is a set of fundamental solution. How should I find $p(x), q(x)$ and $r(x)$?
Thanks for your time.