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**Oblivionwarrior** Determine the recurrence relation for the coefficients of the power series solutions, and find explicitly the …first four non-zero terms in each of the two linear independent series solutions to: $\displaystyle y^{''} +xy^{'} +2y = 0 $ about Xo=0

Let $\displaystyle y = \sum_{n=0}^{\infty}a_n x^n$ then it means,

$\displaystyle \sum_{n=2}^{\infty} n(n-1)a_nx^{n-2} + x\sum_{n=1}^{\infty} na_nx^{n-1} + 2\sum_{n=0}^{\infty} a_n x_n = 0$

This becomes,

$\displaystyle \sum_{n=2}^{\infty}n(n-1)a_nx^{n-2} + \sum_{n=1}^{\infty} na_nx^n + 2\sum_{n=0}^{\infty}a_nx^n = 0$

Change index,

$\displaystyle \sum_{n=0}^{\infty} (n+2)(n+1)a_{n+2}x^n + \sum_{n=1}^{\infty}na_nx^n + 2\sum_{n=0}^{\infty}a_nx^n = 0$

Evaluate the first and third sum at $\displaystyle n=0$ and combine,

$\displaystyle 2a_2 + 2a_0 + \sum_{n=1}^{\infty} [(n+2)(n+1)a_{n+2} + (n+2)a_n]x^n = 0$

But this is the same as writing,

$\displaystyle \sum_{n=0}^{\infty} [(n+2)(n+1)a_{n+2} + (n+2)a_n]x^n = 0$

The recurrence relation is therefore, $\displaystyle (n+2)(n+1)a_{n+2} + (n+2)a_n = 0 \implies a_{n+2} = -\frac{a_n}{n+1} \text{ for }n\geq 0$