How can I find all the derivatives of the function?

Question

My function is the following

$[f(x)]^2 = e^x \int_{1}^{x} t f(t) dt$ While $x \epsilon \mathbb{R}$

And I need to find all the differentiable functions for f All I can think of is using the first fundemental theorem in calculus to get rid of the interval.

Answer 1

First divide both sides by $e^x$, then differentiate; divide by $f(x)$ (assuming $f$ is not identically $0$), and solve the resulting differential equation with the initial condition $f(1)=0$.