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:

"image"

step-by-step explanation:

in most of these problems, the notation is that the point with a "prime" is the image of the point without. that is, b' is the image of b.

here, there is no dilation or reflection. there is simply translation (with no direction shown). which figure you call the "preimage" and which one you call the "image" is up to you. as a matter of convention, the naming is as described above.