Question

Find the power series solution of the differential equation

𝑦``+𝑥𝑦`+(𝑥2+2)𝑦=0

Answer 1

Solution

Let

$y\left(x\right)=\sum_{n=0}^{\infty}{a_nx^n}$

$\frac{dy}{dx}=\sum_{n=1}^{\infty}{na_nx^{n-1}}=\sum_{n=0}^{\infty}{\left(n+1\right)a_{n+1}x^n}$

$\frac{d^2y}{dx^2}=\sum_{n=0}^{\infty}{n\left(n+1\right)a_{n+1}x^{n-1}}=\sum_{n=0}^{\infty}{\left(n+1\right)\left(n+2\right)a_{n+2}x^n}$

Substitution into equation:

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

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

$2a_2+6a_3x+2a_0+3a_1x+\sum_{n=2}^{\infty}{\left(n+1\right)\left(n+2\right)a_{n+2}x^n}+\sum_{n=2}^{\infty}{na_nx^n}+\sum_{n=2}^{\infty}{a_{n-2}x^n}+2\sum_{n=2}^{\infty}{a_nx^n}=0$

Coefficient near xⁿ are equal to zero.

n=0: 2a₂ + 2a₀=0   =>  a₂ = -a₀ 

n=1: 6a₃+3a₁=0   =>  a₃=-a₁ /2

n>1: (n+1)(n+2)a_n+2+na_n+a_n-2+2a_n=0 =>  (n+1)(n+2)a_n+2+(n+2)a_n+a_n-2=0

a_n+2 =-a_n/(n+1)-a_n-2/[(n+1) (n+2)]

Therefore for arbitrary a₀ and a₁ and last recurrent formula solution is

$y\left(x\right)=a_0+a_1x\ -a_0x^2-\frac{a_1}{2}x^3+\sum_{n=2}^{\infty}{a_{n+2}x^{n+2}}$