I have a non-linear integral equation that I'd like to solve with Mathematica: enter image description here

g[y_] := y^2;
h[y_] := y^2/2 + c;
func=(g[y] f[y]/Sqrt[h[y]^2 + f[y]^2]);
  • 1
    $\begingroup$ It would be important to know if numerical solution is fine, or if you're attempting to find something analytically. I guess the latter, or do you have specific values for c? $\endgroup$
    – Lukas
    Aug 19 '16 at 9:18

You should take the derivative with respect to $x$ of both sides. Then you obtain the differential equation $$ f'(x) = \frac{g(x) f(x)}{\sqrt{h(x)^2+f(x)^2}}$$ and $f(0)=0$. The latter can be solved using

g[x_] := x^2;
h[x_] := x^2/2 + c;
DSolve[{f'[x] == g[x] f[x]/Sqrt[h[x]^2 + f[x]^2], f[0]==0}, f[x], x]

It turns out that Mathematica is not able to solve the resulting differential equation.

Looking at the numerical solution (setting a value for $c$ and replacing DSolve by NDSolve), I believe that the (only) solution to your problem is $$ f(x) \equiv 0.$$

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    $\begingroup$ By repeated differentiation, it is clear that f has zero derivatives of all orders at 0 $\endgroup$
    – mikado
    Aug 19 '16 at 14:37
  • $\begingroup$ @mikado: interesting observation. Still there could be a solution that is not analytic as $x=0$... $\endgroup$
    – Fabian
    Aug 19 '16 at 20:05

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