How to solve second order non-linear ODE $$y'^2+y y''+\frac{2}{x} y y' -0.1 y^2=0$$ subject to $y'(1)=0$ and $y(1)=1$ over the interval $0 < x \le 1$.
I turned the equation to a PDE $y'^2+y y''+\frac{2}{x} y y' -0.1 y^2=y'_t$. I was trying to find the steady state solution when $t \to \infty$, which is the solution to the ODE. I used an explicit finite difference scheme in MATLAB. But it doesn't seem to give the right solution.
I have problem implementing boundary conditions in the MATLAB.
I will be grateful if you help me solve this. Also mentioning any other numerical method will be great. Thanks.