Suppose $f,g:[0,\infty)\to [0,\infty)$ are two continuous functions satisfying $$f(x)-g(x) = \int_0^x\frac{g(y)-f(y)}{g(y)f(y)}\;dy$$ for every $x\in[0,\infty)$. How to show that $f(x)=g(x)$ for all $x\in[0,\infty)$?
This question comes from a problem of proving there is at most one nonnegative solution (if exists) for integral equations $$f(x)=a(x)+\int_0^x\frac{1}{f(y)}\;dy$$ where $a(x)$ is a given continuous function with $a(0)\geq0$.
Let $h(x):=f(x)-g(x)$. Then $h'(x)=-\frac{h(x)}{g(x)f(x)}$ so $$ h(x)=C\exp\left(-\int_0^x \frac{1}{g(y)f(y)}\,dy\right). $$