Solve system of non-linear differential equations

I have a large system of non-linear differential equations of the form:

$x_{i}''(y) = f_{i}(x_{1}, x_{2},\ldots,x_{n},x_{1}', x_{2}',\ldots,x_{n}', x_{1}'', x_{2}'',\ldots,x_{i-1}'',x_{i+1}'',\ldots,x_{n}''), \quad i=1,\ldots,n$

which I need to solve numerically. However, the presence of the second-order derivatives in the right-hand side poses a problem. I know I can try and work out how to convert the system to a proper form, but given the amount and complexity of the actual equations, it might prove extremely hard/impossible.

What alternatives do I have? I need only a numerical solution, so I was thinking of perhaps using Newton's method to solve the non-linear system and find the values of $x_{i}''(y)$, but perhaps there are other methods for such problems which I don't know of?

• Is there a reason that you chose $x(y)$ instead of the more traditional $y(x)$ or $x(t)$? – LutzL Feb 15 '15 at 19:15