Second order ODE with mixed derivatives

Question

I'm currently trying to solve a problem which is attached to a differential equation of the form

$$a(x,y(x))y''(x)+b(x,y(x))y''(x)y'(x)+c(x,y(x))y'(x)+d(x,y(x))=0$$

The term $a,b,c,d$ are all in about the same range of values. I'm trying to find an approximation or numerical solution, so I wanted to ask if anybody has some advice how to approach a problem of this kind? Is there maybe some numerical algorithm for equations of this form?

Answer 1

There is nothing special about this. Bring the equation to explicit form $$ y''=F(x,y,y')=-\frac{c(x,y)y'+d(x,y)}{a(x,y)+b(x,y)y'} $$ transform into a first order system and apply the numerical method of your choice.