Having the following set of equations;
\begin{align} f_1(x) &= \frac{1}{f_2(x)}\\ f_1(x) &= \int^{x_1(x)}_0 \phi(t) dt\\ f_2(x) &= \int^{x_2(x)}_0 \phi(t) dt, \end{align}
I'd like to solve for $\phi$. The functions $x_1(x)$ and $x_2(x)$ are known, continuous and strictly monotonic. I could use the fundamental theorem of calculus to achieve
\begin{align} \frac{df_1}{dx} &= \phi(x_1) \frac{dx_1}{dx} &\frac{df_2}{dx} &= \phi(x_2) \frac{dx_2}{dx}, \end{align}
and then take the derivate of the first equation like so
\begin{equation} \frac{df_1}{dx} = -\frac{1}{f_2^2} \frac{df_2}{dx}. \end{equation}
But I still cannot get rid of the $f_2$ and don't know if that makes the situation any better. Is this a well-known type of mathematical problem? And is there a way to solve this or similar problems computationally, e.g. with root-finding or the finite difference method?
Many thanks in advance.