How to calculate the solutions of indicial equation of $$2xy''-xy'+(1+x)y=0?$$
My working: After applying the Frobenius method, the lowest power of $x$ in terms of $r$ comes out to be $r-1$ (putting $n=0$), and the corresponding indicial equation turns out to be $2r(r-1)=0$ which gives $r=0$ and $r=1$ as the solutions of the indicial equation.
- Are the method and the values correct?
- Also can anyone clarify that the indicial equation is the equation corresponding to the lowest power of $x$ in terms of $r$ (which is $r-1$ in this case) or is it the equation corresponding to the $r$th power of $x$ (if by definition)?