Let y=∑anxn, then y'=∑nanxn-1, y"=∑n(n-1)anxn-2 for integer n≥0.
16y=16a0+16a1x+16a2x2+16a3x3+16a4x4+16a5x5+...
-7xy'=-7a1x-14a2x2-21a3x3-28a4x4-35a5x5-...
x2y"=2a2x2+6a3x3+12a4x4+20a5x5+...
The DE LHS becomes:
16a0+9a1x+4a2x2+a3x3+∑(n-4)2anxn for n≥4 (x4 has a coefficient of 0).
The DE RHS expands to:
x4(x-x2/2+x3/3-x4/4+...) so the lowest power of x is x5. Therefore a0 to a4 are zero.
x5 has a coefficient of 1, x6 a coefficient of 1/2, x7 a coefficient of 1/3, xn a coefficient of 1/(n-4).
Equating LHS and RHS coefficients:
(n-4)2an=1/(n-4), from which an=1/(n-4)3.
y=∑xn/(n-4)3 for n≥5, y=x5+x6/8+x7/27+x8/64+...