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Differential Equations

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Moderate

Series Solution with Nonzero Center

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Lisa is curious about the behavior of the solutions for the equation $y''+(2x+1)y'+2y=0$ near $x=2$, which is an ordinary point.

She thinks there exists a solution of the form $y=\sum_{n=0}^{\infty}a_n(x-2)^n$.

The recurrence relation Lisa will find could be:

A

$a_{n+2}=-\cfrac{1}{n+2}a_{n+1}-\cfrac{2}{n+2}a_n,\ n\ge0$

B

$a_{n+2}=-\cfrac{2x+1}{n+2}a_{n+1}-\cfrac{2}{(n+2)(n+1)}a_n,\ n\ge 0$

C

$a_{n+2}=\cfrac{3}{n+2}a_{n+1}-\cfrac{2}{n+2}a_n,\ n\ge0$

D

$a_{n+2}=-\cfrac{5}{n+2}a_{n+1}-\cfrac{2}{n+2}a_n,\ n\ge0$