I am trying to solve this equation: $$ \frac{d^2y}{dx^2}-2x\frac{dy}{dx}+2 \alpha y=0$$
I am using Frobenius method, I rewrote the equation to: $$ \frac{d^2y}{dx^2}-2\frac{x^2}{x}\frac{dy}{dx}+2 \frac{x^2}{x^2}\alpha y=0$$ so it has the form of: $$ \frac{d^2y}{dx^2}-\frac{g_1(x)}{(x-x_0)}\frac{dy}{dx}+\frac{g_2(x)}{(x-x_0)^2} \alpha y=0$$
And I can find the roots of the indicial eqaution: $$\rho(\rho-1)+g_1(x_0) \rho+g_2(x)=0$$ So $x_0=0$ and the indicial equation is: $$\rho(\rho-1)+g_1(0) \rho+g_2(0)=\rho(\rho-1)+0 \dot \,\rho+=\rho(\rho-1)=0$$ $$p_1=0, p_2=1$$ Before I continue to propose a solution in the form of: $$ y= x^\rho \sum^{\infty}_{n=0}a_n x^n$$ I want to know if what I am doing is correct.
Thanks in advance.