# Solution to non-linear OIDE

How do I go about solving this equation?

$\frac{\partial F(r,y)}{\partial r} = Q(r,y) - P(r,y) F(r,y) - R(r,y)F(r,y)\int_0^\infty dy'S(r,y') F(r,y')$

with the initial condition that $F(r=0,y) = 0 \ \ \forall y$

The Q's, P's R's and S's are arbitrary but well defined functions.

The above OIDE was derived whilst calculating the dark matter density using the Boltzmann equation. I have looked up various methods of solving this, both analytically and numerically. The methods which I have attempted are the Laplace transform methods and Volterra method. However, the non-linear component in the last term of the RHS is proving to be a bit difficult.

Many thanks!

• I figured out a way to solve this numerically. First, divide both sides by R(r,y)F(r,y). Then, differentiation wrt dy gets rid of the integral and from there on, proceed with the resulting differential equation. – conformist_anomaly Sep 23 '14 at 19:24