There are a lot of questions like this on MSE as well as online resources on the subject, but a) the MSE questions are either unanswered or correspond to systems substantially different from this one, and b) the online resources I've found (e.g., here) always jump directly to phase plane analysis and/or numerical methods. The system I want to solve is
$$V' = aV^2 + bV - U + c \\ U' = dV -kU $$
Initial conditions are known (let them be $v_0$, $u_0$). So, it's linear except for the $aV^2$ term. Can it be solved analytically? I've thought about setting $X = V^2$ and turning it into a 3-D linear system that way, but I'm not sure if that is the correct approach. I've thought about Laplace transforms too but it looks like applying them to $aV^2$ would be extremely complicated.
Lastly a solution for the simpler case where $c=0$ would be acceptable to me, I can probably figure out how to extend it to the $c\neq0$ case on my own.
Edit: I tried the following, with $X(0) = v_0^2$:
$$X' = 2V\\ V' = aX + bV - U + c \\ U' = dV -kU $$
It failed pretty dramatically.
Since the system of two ODEs can be transformed to an Abel's differential equation of first kind, there is no known closed form for the solutions in the general case.
However, closed forms might exist in some particular cases (i.e. for particular values of the parameters $a,b,c,d,k$ ).