Solving nonlinear integral equation

Question

Are there some known techniques for solving $h(x)=f(x)\int_0^xf(t)dt$ for $f(x)$? Are there closed form solutions?

Answer 1

Let $F(x)=\int_0^xf(t)\,dt$. Then $F'=f$, so that the integral equation transforms into the differential equation $$ F'F=h,\quad F(0)=0. $$ Integrating we get $$ \frac12 F^2=\int_0^xh(t)\,dt\implies F(x)=\sqrt{2\int_0^xh(t)\,dt} $$ and $$ f(x)=F'(x)=\frac{1}{\sqrt2}\frac{h(x)}{\sqrt{\int_0^xh(t)\,dt}}. $$