# Integrating a nonlinear ordinary differential equation

I am solving an equation of the form

$(*)$ $0 = a(f) (\partial_rf)^2 + b(f) (\partial_rf) + c(f),$

where $f$ is a real function of $r\in \mathbb{R}$, and $a,b,c$ are real functions of $f$. The boundary condition $f=f_0$ at $r=0$. Right now, I am solving $(*)$ by finite differencing $(*)$ and then applying a relaxation method.

I want to solve $(*)$ using an explicit method, such as a Runge-Kutta type method. The only idea I have thought of so far is treating $(*)$ as an algebraic equation for $\partial_rf$, and then solving the resulting linear ODE using a Runge-Kutta solver. I find that procedure to be very inelegant though, and I'd like to know if there there is a body of algorithms that can solve (at least) specific kinds of nonlinear ODE.

I recognize there (may be?) issues with uniqueness of the solution(s) of $(*)$; I would appreciate any insights on how to numerically select a specific solution as well.

• Just to clarify: in $(*)$, if the discriminant $b^2-4ac>0$, could I in principle could reduce the equation to a first order ODE which I could solve using an explicit method? I am also concerned about the uniqueness of the solution of $(*)$. – Justin Mar 4 '18 at 23:12