Functional equation for f on the half-line

Let $f : [0, \infty) \to [0, \infty)$ be a differentiable function with $f(x) > 0$ for all $x > 0$, and $f(0) = 0$. Further, $f$ satisfies $(f(x))^2 = \int_{0}^{x} ((f(t))^2 + f(t)) \, dt, \quad x > 0$. Then which one of the following options is correct?