I have formulated a function $f(x)$ as the solution to an ordinary differential equation
$$ f'(x) = \phi(f(x),x) \\ f(x_0) = f_0. $$
and also the function $F(x) = \int_0^x f(s)ds$ as
$$ F'(x) = \Phi(F(x),x) \\ F(x_0) = F_0. $$
Is it possible, in general, to prove that the two formulations are equivalent (for differentiable functions)?
Actually, you have to consider the functions $\phi(x,y)$ and $\Phi(x,y)$ as bivariate, such that $$ \phi(x,f(x)) = f'(x) = F''(x) = \frac{\mathrm{d}}{\mathrm{d}x}\Phi(x,F(x)) = \Phi_x(x,F(x))+\Phi_y(x,F(x))f(x), $$ where the indices on $\Phi$ denote partial derivatives, with the boundary condition $\Phi(x_0,F_0) = f_0$. Recalling that $f(x) = \Phi(x,F(x))$ and setting $y = F(x)$, one gets the following partial differential equation : $$ \Phi_x(x,y) + \Phi_y(x,y)\Phi(x,y) = \phi(x,\Phi(x,y)) $$ which is a very complicated problem in general, because of its non-linearity and the composition on the right-hand side.