## Hint for Problem 1, Ass't 5

The problem is to show (a) that ln(x) is not a ratio of
two polynomials, i.e. that q(x)ln(x) + p(x) = 0 implies
that q(x) and p(x) are both identically zero, and (b)
that a[m](x) ln(x)^m + ... + a[0](x) = 0 likewise
implies that the a[k]'s are all zero.

The key idea is to differentiate q(x)ln(x) + p(x) enough
times so that the coefficient of ln(x) is zero. Then you
can work your way back using polynomial solutions of linear
differential equations.

This idea works for part (b) of Question (1) as well. Note
also that part (a) is contained in part (b); if you can prove
part (b), you've proved part (a).