Nonlinear ODE of second order with Boundary Conditions defined. The problem is:
$y''(x)-a\cdot(y(x))^2=0, a>0$
S.t. $ y(0)=b,
\lim_{x\to\infty } y'(x)=0$
That problem results from a catalyst which has a chemical reaction of second order occuring within it - the book Transport Phenomena of Bird at. al. contains that question. Someone can give me a tip how to proceed to solve that nonlinear ODE? That is the first time that I found this kind of problem.

You start by setting $y'=z$ in order to come up with a first order separable differential equation. 
(Sorry for the short answer but I thought since you asked for a tip I should not write the whole solution).
