Use power series methods to solve the differential equation

Use power series methods to solve the differential equation y \'\' + (x 1) y = 0 at the point x\'= 0. It suffices to give the first six nonzero terms of the series solution

Solution

Let y = a0+a1x+a2x^2+...+anxⁿ+...

y\' = a1+2a2x+...+nanx^n-1+...

y\" = 2a2+6a3x+...+n(n-1) anx^n-2+...

y \'\' + (x 1) y = 0

becomes

2a2+6a3x+...+n(n-1) anx^n-2+...

a0x+a1x²+a2x³+...+nanxⁿ⁺¹+...

-(+a0+a1x+a2x^2+...+anxⁿ+...)=0

Equate x^n to 0

2a2-a0 =0

a2 = 1/2 a0

6a3+a0-a1 =0

a3 = a1-a0

12a4+a1-a2 =0

12a4+a1-a0/2 =0

20a5+a2-a3 =0

20a5+a0/2-(a1-a0) =0

20a5 = a1+a0/2

(n+2)(n+1) a_n+2+(n-1)a_n-1-a_n=0