A system of DFQ's... (non-linear)

Hello!

I've got the following problem to solve:

Prove that every solution of:

{dy/dx = x^3(cosy) + x^4(sinz) + x^5

{dz/dx = x^6(sinz) + x^7(cosy) + x^8

is defined on all of R.

I have absolutely no idea how to solve this problem... I've only learned of linear systems... Can I somehow transform it into one? Is it even nescesary? I've read somewhere that it's enough to prove that both of the functions in the equations satisfy Lipschitz law (the first in respect to y, and the second in respect to z), but I'm not sure about it...