I cannot understand which one to use when solving differential equations by using power series solutions. For example in this question:
Consider the following differential equation for $\alpha \in > \mathbb{R}$: $xu''+(x-1)u' -\alpha u=0.$ Determine the two values of $\alpha$ so that the solutions to this differential equation around $x = 0$ have no logarithmic part.
My solution book uses $\sum_{n=0}^\infty a_nx^{n+r}$. Why is that? And for the following question:
Find the two linearly independent power series solutions of
$(1+x^2)y''+2xy'-2y = 0,$
about the point $x = 0.$ In each of the solutions determine the general term and the radius of convergence of the series.
My solution book uses $\sum_{n=0}^\infty a_nx^{n}$. You don't need to solve the questions but I don't understand and can't choose between them.
The solution with $x^{n+r}$ is not a power series solution as there is no assumption that $r$ is an positive integer. In fact, $r$ could be complex! This is the method of Frobenius which provides solutions to differential equations with regular singular points. You should read about the method of Frobenius, singular points and regular singular points. This will help clarify it for you. Long story short, the solution is a power series at an ordinary point, but otherwise it's tricky.