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Differential Equations

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Recurrence Relation for a Series about Zero

DIFFEQ-2RSJNL

Consider the equation $y''-xy'-y=0$.

If we seek a power series solution $y=\sum_{n=0}^{\infty}a_n(x-c)^n$ with center $c=0$, the recurrence relation is given by:

A

$a_{n+2}=\frac{1}{n+2}a_n, n=0,1,2,\ldots$

B

$a_{2k+1}=0, k\in\mathbb{Z}_{+}$$, $$a_{2k}=\frac{1}{2k}a_{2k-2}, k\in\mathbb{Z}_{+}$

C

$a_{n+2}=\frac{x}{n+2}a_{n+1}+\frac{1}{(n+2)(n+1)}a_n$

D

$a_0=0, a_2=0,\ a_{n+2}=\frac{1}{n+2}a_n, n\ge 1$