Frobenius method, reduction of order technique
So this is the problem:
xy'' + y'- xy = 0
Find the first nonzero terms in a Frobenius series solution of the given differential equation. Then use the reduction of order technique to find the logarithmic term and the first three nonzero terms in a second linearly independent solutions.
I did the first part:
n(n+1)cn+1xn + (n+1)cn+1xn - cn-1xn= 0
[n(n+1) + (n+1)]cn+1 = cn-1
cn+1 = cn-1/ n(n+1) + (n+1)
c1 = 0
c2 = c0/22
c3 = c1/32 = 0
c4 = c2/22*42
y1(x) = 1 + x2/22 + x4/22*42 + x6/22*42*62 + x8/22*42*62*82 + ...
OK, and now to find y2, I'm kind of stuck. Since in the indicial equation, the roots are both r=0, so the equation I should use is: y2 = y1*ln(x) + xr1+1 bnxn.
The solution manual of my text book starts off with this:
y2 = y1 x-1 * (1 + x2/22 + x4/22*42 + x6/22*42*62 + x8/22*42*62*82)-2 dx
... I have NO clue where that came from. Someone please explain how to solve this? I'm studying for an exam and an explanation would be immensely helpful.