Let y=∑anxn for n≥0.
The first few terms are therefore:
y=a0+a1x+a2x2+a3x3+... (general term anxn),
y'=a1+2a2x+3a3x2+... (general term nanxn-1),
y"=2a2+6a3x+12a4x2+20a5x3... (general term n(n-1)anxn-2),
x2y'=a1x2+2a2x3+3a3x4+... (general term nanxn+1),
xy=a0x+a1x2+a2x3+a3x4+... (general term anxn+1).
Add the last three equations together:
2a2+(a0+6a3)x+(2a1+12a4)x2+(3a2+20a5)x3+...
(general term (nan-1+(n+1)(n+2)an+2)xn)=0
Each coefficient of x must be zero for the whole series to sum to zero for general x.
Therefore a2=0. a0 and a1 are unknown constants. Let a0=A and a1=B.
6a3=-A, a3=-A/6; a4=-B/6; a5=0; an+2=-nan-1/[(n+1)(n+2)].
This makes y=A+Bx-Ax3/6-Bx4/6+Ax6/45+5Bx7/252-...
Note that the coefficient for x2, x5, x8, etc., are zero.