y'' - xy' + 2y = cosx

Question

Solve the following non-homogeneous ODE using a power series expansion centered at 0. Listing the first four nontrivial terms for y1 and y2.

Answer 1

Let y(0) = A and y'(0) = B

if y'' -xy' + 2y = cos(x) then y'' = xy' - 2y + cos(x) therefore;

y''(0) = (0)B - 2A + cos(0) = 1 - 2A

Next, y''' = y' + xy'' - 2y' - sin(x) so, y'''(0) = B + (0)(1 - 2A) - 2B - 0 = -B

Therefore; y(x) = y(0) + y'(0)x + y''(0)x²/2! + y'''(0)x³/3! ........

y(x) = A + Bx + (1-2A)x²/2! + (-B)x³/3! (first 4 terms)