Given this boundary value problem
$y'' = (x^2(y')^2 - 9y^2 + 4x^6)/x^5, \quad 1 \leq x \leq 2, \qquad (1)\\ y(1) = 0, \; y(2) = \ln 256$
I have to solve the problem using finite differences, for 21 points (difference = 0.05), and use Newton's method to solve the nonlinear system generated by the problem discretization.
Using the $y'$ and $y''$ substitutions by finite differences in the equation 1, I've got that
$ - (x_j^2/4)y_{j+1}^2 + (9 h^2)y_j^2 - (x_j^2/4) y_{j-1}^2 + (x_j^5)y_{j+1} + (x_j^2/2) y_{j+1} y_{j-1} + (x_j^5)y_{j-1} - (2x_j^5)y_j - 4x_j^6h^2 = 0 $
I'd know how to solve if it was linear, but I still don't know how to solve this by Newton's method.