I have the following differential equation $$4xy''+2(x+1)y'+y=0$$
I need to use the Frobenius method to determine the indicial equation, recurrance relation and the roots of the indicial equation. So far I have
$$\sum_{n=0}^\infty 4a_n(n+r)(n+r-1)x^{n+r-1} + \sum_{n=0}^\infty 2a_n(n+r)x^{n+r} +\sum_{n=0}^\infty 2a_n(n+r)x^{n+r-1}+\sum_{n=0}^\infty a_nx^{n+r}$$
I am stuck on writing all the powers of $x$ as $x^{n+r}$
you must change the variables $n$ in each summation for $k$. For instance,
$$\sum_{k=0}^\infty \left\lbrace 4a_k(k+r)(k+r-1)x^{k+r-1} \right\rbrace + \sum_{k=1}^\infty \left\lbrace 2a_{k-1}(k+r-1)x^{k+r-1} \right\rbrace +$$ $$\sum_{k=0}^\infty \left\lbrace 2a_k(k+r)(k+r)x^{k+r-1} \right\rbrace + \sum_{k=1}^\infty \left\lbrace a_{k-1}x^{k+r-1} \right\rbrace = 0$$
Both changes were: $n_0^\infty = k_1^\infty-1$
Can you finish the rest?
