Finding the Series Solution

Question

Find the series solution(s) centered at 0 of the differential equation 2xy′′ + (1 + x) y′ + y = 0.

Answer 1

No initial conditions were given? So, using the general formula y = ∑ a_nx^(m+n)

where m = the numbers to be solved so that will allow a_o be non-zero. m is found to be 0 and 1/2

The recursive formula turns out to be a_n = a_n-1/(2m+2n-1)

so if y_o = solution for m = 0 and y_1/2 is the solution for m = 1/2 and allowing a_o = 1 we find that

y = Ay_o + By_1/2 = A(1 - x/2 + x²/3 - x³/15 + x⁴/105 - x⁵/945 + x⁶/10,395 ......) +

Bx^(1/2)(1 - x/2 + x²/8 - x³/48 + x⁴/384 - x⁵/3840 + x⁶/46,080 ........)

A & B are arbitrary constants