A general approach to quasilinearization of nonlinear ODEs is described in the article linked below.

http://arxiv.org/pdf/physics/0102041

y''(x)=y^3(x)-y(x)y'(x) .Replace y3(n+1) = y3(n) + 3y2(n)(y(n+1) - y(n)) = 3y2(n)y(n+1) - 2y3(n)  and replace y(n+1)y'(n+1) = y(n)y'(n) + (y(n+1)-y(n))y'(n) + (y'(n+1) - y'(n))y(n)

= y(n+1)y'(n) + y'(n+1)y(n) -y'(n)y(n).

Here n indicates iteration where assume y(0) = (2-x)/2 +(x-1)/3

A general approach to the quasilinearization technique of nonlinear ODEs is described in the article linked below.