use the power series method to solve the given initial-value problem. (format your final answer as an elementary function.)
(x − 1)y'' − xy' + y = 0, y(0) = −3, y'(0) = 2

Notice that , and .

Substitute these series into the ODE:

Shift the indices to get each series to include a term.

Remove the first term from both series starting at to get all the series starting on the same index :

The coefficients are given recursively by

Let's see if we can find a pattern to these coefficients.

and so on, suggesting that

which is also consistent with . However,

but we can adjust for this easily:

Now all the terms following resemble an exponential series:

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step-by-step explanation:

because i said it is