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Consider an ODE of the form $y'' + P(x)y' + Q(x)y = 0$ where $(x-x_0)P(x)$ and $(x-x_0)^2 Q(x)$ are analytic functions on $x_0$. The Frobenius Theorem asserts that there exist at least one solution of the form: $$ (x - x_0)^r \cdot \sum_{k = 0}^\infty a_k (x - x_0)^k $$ where $r$ is an incognit, there is a formula for finding the $r$?

In some books appears, when $x_0 = 0$ that changes something? If this is the case, the formula is given by $$ r(r-1) + p_0 r + q_0 = 0 $$ where $q_0, p_0$ are the first terms of $$ \begin{align*} P(x) &= (x-x_0)^{-1} \cdot \sum_{k = 0}^\infty p_k (x-x_0)^k \\ Q(x) &= (x-x_0)^{-2} \cdot \sum_{k = 0}^\infty q_k (x-x_0)^k \end{align*} $$ Thanks, I only want to know that.

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