Is there any solution formula for a differential equation like this
$$y'(x)=f(x)y(x)+g(x)y(ax)\quad\text{ where }a>1.$$
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.
If you change the independent variable $x = e^t$, the equation becomes a pretty typical delay differential equation. If you have suitable initial condition, the problem reduces to a sequence of initial value problems for ODE. It is not actually necessary to change the variable; working in terms of $x$ you can do the same things, only the steps will be nonuniform in size: $[a^k,a^{k+1}]$.
Getting an exact formula for solution, when the DDE has variable coefficients, would take a miraculous coincidence. Don't expect it.