Considering the following equation $$ x^2 y''(x)+a(x)y'(x)+b(x)y(x)=0$$ with $y(0)=1$ and $y'(0)=1$.
How could I find numerically a solution?
Here $a(x=0)$ and $b(x=0)$ are 2 complex constants.
What I tried is, to start at $x=\epsilon$ from which I'm getting a solution and I tried to reduce the value of $\epsilon$ to check the convergence of the solution. The solution is strongly dependent on the value of $\epsilon$.
I've seen similar problems, known as singular perturbation boundary layer problem. Any idea on how to solve these type of equations?