= Solution
The homogeneous <linear differential equation> can be written in <divergence form>:
$$
(x^{n+1}u')'=0.
$$
One integration gives $u'=C x^{-n-1}$. Therefore the two cases are
$$
\boxed{u_h(x)=A+B x^{-n}\quad(n\ne0),\qquad u_h(x)=A+B\log x\quad(n=0).}
$$
In each case the displayed functions have <linear independence> throughout the interval. Imposing both homogeneous <Dirichlet boundary conditions> forces $A=B=0$: for $n\ne0$, $2^n\ne1$, and for $n=0$, $\log2\ne0$. This also establishes uniqueness for the inhomogeneous <boundary value problem>.
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