Is there any solution formula for a differential equation like this $$ y'(x)=f(x)y(x)+g(x)y(ax)\qquad \text{where}\qquad a\gt1. $$ I have a differential equation a little more complicated than this. The basic difficulty for me is that, I don't know how to deal with $y(ax)$.
Thank you.
Even for the simplest equation of this sort, $y'=y(ax)$, there is no simple formula. Substitute a power series at $0$, and you obtain a series which is divergent when $|a|>1$. This shows that there is no solution analytic at $0$. (If $|a|\leq 1$ this series is convergent everywhere, but still no formula is known simpler than the series itself.) On the other hand, you can reduce to a differential-difference equation, as Anthony Quas recommends, and if $f$ and $g$ are constants, this differential-difference (DD) equation can be solved with Fourier or Laplace transform, depending on your assumptions about $y$. Of course the DD equation always has infinitely many simple discontinuous solutions, depending on an arbitrary function, but probably this is not what you are looking for.
