Question

Find a power series solution of the differential equation y" +x^2 y'=0 about the ordinary point x =0.

Answer 1

Answer:

Step-by-step explanation:

$Let y=a0+a1x+a2x^2+...\\y'= a1+2a2x +3a3x^2+...$

$y"=2a2+3(2)a3x+....+n(n-1)anx^n-2 +...\\y"+x^2y'=0\\2a2x^2+3(2)a3x^3+....+n(n-1)anx^n +a1+2a2x+....+nanx^{n-1} +...=0\\\\a2=0 :\\a1=0\\\\(n-1) na_n +(n+1)a_{n+1} =0$

Hence recurring formula is

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

with first term arbitrary and second and third term =0