Solving the Differential equation y′′− x^3y = 0

Question

Solve the differential equation y′′− x^3y = 0 using power series centered at 0.

Answer 1

You did not state the initial conditions. Let's assume y(0) = A and y'(0) = B

This is a strange equation & the series solution will go on forever but the first few terms would be

y = A + Bx + (A/20)x⁵ + (B/30)x⁶ + (A/1,800)x¹⁰ + (B/3,300)x¹¹ + ................

if x, A, & B are relatively small, the series converges quickly!