Find two linearly independent power series solutions for the DE

Question

Find two linearly independent power series solutions for the DE, giving three nonzero terms for each

y''+y'+xy=0

Answer 1

You didn't specify any initial conditions for the problem! So, let's assume that y(0) = A and y'(0) = B

By rearranging, we have

y'' = - y' - xy and y''(0) = -B or f''(0) = -B

y''' = -y'' - y - xy' y'''(0) = B - A f'''(0) = B - A etc

y^iv = -y''' - 2y' - xy'' y^iv = A - 3B

y^v = -yiv - 3y'' - xy''' y^v = 6B + A

Tayor's Formula says that y = f(x) = f(0) + f'(0)x + f''(0)x2/2! + f'''(0)x3/3! + .............

Just plug in the values for each to obtain the 2 independent equations!

Hope this helps!