Proceed as in Example 4 in Section 6.1 and find a power series solution y = [infinity] n = 0 cnxn of the given linear first-order differential equation. (Give your answer in terms of c0.) y' = xy

Let y be a solution to the given differential equation,

where

Substituting these series into the DE gives

Then the coefficients in the series solution are governed by the recurrence,

We have

so it follows that , while

and so on, with the general n-th coefficient being

Then the power series solution is

but this doesn't tell the whole story because it doesn't capture the odd-index-is-zero case.

More concisely: let n = 2k for integers k ≥ 0. Then

and as a bonus, it's easier to get an exact solution for this DE,

50.6

step-by-step explanation: