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]);
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1 Answer
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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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1$\begingroup$ By repeated differentiation, it is clear that
fhas zero derivatives of all orders at 0 $\endgroup$– mikadoCommented Aug 19, 2016 at 14:37 -
$\begingroup$ @mikado: interesting observation. Still there could be a solution that is not analytic as $x=0$... $\endgroup$– FabianCommented Aug 19, 2016 at 20:05
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