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.

  • 3
    $\begingroup$ If you write $t=\log x$, then this turns into a delay differential equation. These are known to be hard. $\endgroup$ – Anthony Quas May 28 '14 at 4:31
  • $\begingroup$ I put the maths into LaTeX. $\endgroup$ – David Roberts May 28 '14 at 4:56
  • $\begingroup$ why don't you put the example that you have ?? $\endgroup$ – ss1729 Mar 29 '15 at 21:12

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.

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