Can't seem to get a recurrence relation when applying the Frobenius Method to the following ODE.

Question

$$x^2y'' -2xy' + 2y = 0 $$ Substituting in $ y = \sum_{i = 0}^\infty a_ix^i$ to obtain: $$ \sum_{i = 2}^\infty i(i-1)a_ix^i -2\sum_{i = 1}^\infty ia_ix^i +2\sum_{i = 0}^\infty a_ix^i = 0. $$ Matching up indexes: $$ \sum_{i = 2}^\infty i(i-1)a_ix^i -2a_1x -2\sum_{i = 2}^\infty ia_ix^i +2a_0 + 2a_1x + 2\sum_{i = 2}^\infty a_ix^i = 0. $$ How do I set up a recurrence relation with this?

Answer 1

If you include terms that are automatically zero, you can start all sums at zero. You get $$ \sum_{i=0}^\infty (i^2-3i+2)a_ix^i=0 $$ which gives exactly two coefficients that can be non-zero.

You are lucky that the indicial equation has integer solutions, otherwise there are no solutions that are simple power series.