I have the following set of (autonomous?) 2nd-order nonlinear ODEs, written as followed: $$f''(x)=af(x)g(x), g''(x) =bf(x)g(x),$$ where $''$ denotes the second derivative with respect to $x$ ($d^2/dx^2$) and $a > 0$ and $b > 0$ are constants. Given the obvious similarity between two equations, and perhaps, their simplicity, I have a hunch that a general, closed-form solution for $f(x)$ and $g(x)$ exists, but I'm not sure where to start. Does anyone have any suggestions on where to begin?
Note 1: The 1st-order variation of this system (where $f''(x)$ and $g''(x)$ are replaced with $f'(x)$ and $g'(x)$, respectively) seems to have a straightforward method for finding a general solution, according to Polyanin and Zaitsev's "Handbook of Ordinary Differential Equations" (2018), example 7.4 (pg 250). There's also a variety of lectures on the 1st-order variation on YouTube.
Note 2: In the 1st-order variation, one usually solves the system by finding the derivative of $f$ with respect to $g$, $df/dg$, via the chain rule and solving from there. However, this 2nd-order system requires a higher-order chain rule (https://en.wikipedia.org/wiki/Chain_rule#Higher_derivatives) in order to solve for $d^2 f/dg^2$, and evaluating some of these derivative terms is non-obvious to me.
Note 3: With the generous advice of @Gonçalo, the equations can be decoupled quite easily since $bf''(x) = ag''(x)$, so $bf(x) = ag(x) + c_1 x + c_2$ and therefore $g''(x) = (ag(x) + c_1 x + c_2)g(x) = ag^2(x) + c_1 x g(x) + c_2 g(x).$ Now, I have a 2nd-order nonlinear ODE to solve. Again pointing to Polyanin and Zaitsev's, this time pointing to section 14.4, they provide some general solutions to a few cases of the ODE $y'' = A_1 x^{n_1} y^{m_1} + A_2 x^{n_2} y^{m_2}$, but I have an extra term that one could call $A_3 x^{n_3} y^{m_3}$. Any suggestions?
Thank you!!