How can I find a solution to the following first order nonlinear (quadratic) ODE?
$$ h(t)=a\dot h(t)\sqrt{b-c \dot h(t)^2} $$
I am solving a physics problem, namely, finding the time it takes for a cylinder to sink in water, and need to find the function $h$.
The water that the cylinder is in can be considered as nonviscous. Therefore I used conservation of energy and the continuity equation to obtain this equation. I think that my physical reasoning is correct although some approximations I made might not be reasonable. I want to verify this by solving this equation and evaluating the time.
In case this raises concerns, I am not interested in obtaining a solution by the work of others without own contribution, I want to know if this ODE is solvable analytically and if so, which methods I should use / learn to solve it. Regarding my knowledge on DEs, aside from special cases like using characteristic polynomials or the harmonic oscillator, I only know separation of variables and variation of constants for inhomogeneous first order linear ODEs.