The problem is solving the ODE $f''(x)+a_1f'(x)+a_2f(x)=g(x)$ with boundary conditions $f(c)=h(c)$, $f(d)=h(d)$, where $c,d$ must be such that $f'(x)\geq b\;\forall x\in[c,d]\subseteq\mathbb{R}$. The constants $a_1,a_2,b\in\mathbb{R}$ are known. The functions $g(x),h(x)$ are smooth and known. The parameters $c,d$ and the function $f(x)$ are not known and should be solved for. It is unlikely that $c,d$ can be solved for uniquely, so the solution would involve a region $D\subseteq\mathbb{R}^2$ for which if $(c,d)\in D$, the ODE can be solved.
What is this type of problem called? What keywords should I search for to find a solution method?
It somewhat resembles differential-algebraic equations, mathematical programming with equilibrium constraints, bilevel optimization.
This arises in a continuous-time game in which I'm trying to find an equilibrium.
This is a linear second order non-homogeneous ODE, if you know g(x) you can try find its general solution and then use the contraints at hand to determine the constants of integration and the parameters c,d.