# nonlinear integral equation

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

g[y_] := y^2;
h[y_] := y^2/2 + c;
func=(g[y] f[y]/Sqrt[h[y]^2 + f[y]^2]);

• 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? Commented Aug 19, 2016 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 $$f(x) \equiv 0.$$

• By repeated differentiation, it is clear that f has zero derivatives of all orders at 0 Commented Aug 19, 2016 at 14:37
• @mikado: interesting observation. Still there could be a solution that is not analytic as $x=0$... Commented Aug 19, 2016 at 20:05