Find the solution for the Integral equation

Question

I have an integral equation like this

$\qquad n(\phi)=\int_0^\sqrt{\phi} f(w)\sqrt{2w+\phi}dw$.

I need to find $f(w)$ analytically. Here $n(\phi)$ is known. Here $\phi$ is a constant.

Answer 1

Suggestion to avoid a common mistake:

Recall Leibniz's Integral Rule:

$$ \frac{d}{dx}\int_{a(x)}^{b(x)} f(x,t)dt = f(x,b(x)) \frac{d b(x)}{dx}-f(x,a(x)) \frac{d a(x)}{dx}+\int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x,t)dt $$

In this case: $$ \frac{dn}{d\phi}=\frac{d}{d\phi}\int_{0}^{\sqrt\phi} f(\omega) \sqrt{2\omega +\phi} d\omega = -f(\sqrt\phi) \sqrt{2\sqrt\phi +\phi} \frac{1}{2\sqrt{\phi}}+\int_{0}^{\sqrt\phi} f(\omega) \frac{1}{\sqrt{2\omega+\phi}} d\omega $$

My suggestion is trying to integrate by parts the last term, hopefully finding something as a function of $n(\phi)$.