Question

Using power series method solve the equation

y''+9y=0

Answer 1

$y = \sum\limits_{n=0}^{+\infty} c_n x^n$ ,

$y'' = \sum\limits_{n=0}^{+\infty} n(n-1)c_n x^{n-2} = \sum\limits_{n=0}^{+\infty} (n+2)(n+1)c_{n+2} x^{n}$

$y'' + 9y = \sum\limits_{n=0}^{+\infty} ((n+2)(n+1)c_{n+2}+9c_n) x^{n} = 0$

$(n+2)(n+1)c_{n+2}+9c_n = 0$

$c_{n+2}=\frac{-9}{(n+2)(n+1)}c_n$

$c_{2n} = \frac{(-9)^n}{(2n)!}c_0$

$c_{2n+1} = \frac{(-9)^n}{(2n+1)!}c_1$

$y=\sum\limits_{n=0}^{+\infty} \frac{(-9)^n}{(2n)!}c_0x^n + \sum\limits_{n=0}^{+\infty} \frac{(-9)^n}{(2n+1)!}c_1 x^n$

$y= c_0\cos{3x} + c_1\sin{3x}$