Solution
= Solution
On the annulus the logarithmic term is allowed, so
$$
u(r)=r^4+C\log r+D.
$$
The condition $u(1)=1$ gives $D=0$, while $u(2)=1$ gives
$$
16+C\log2=1.
$$
Therefore
$$
\boxed{
u(r)=r^4-\frac{15\log r}{\log2}}.
$$
= Solution
On the annulus the logarithmic term is allowed, so
$$
u(r)=r^4+C\log r+D.
$$
The condition $u(1)=1$ gives $D=0$, while $u(2)=1$ gives
$$
16+C\log2=1.
$$
Therefore
$$
\boxed{
u(r)=r^4-\frac{15\log r}{\log2}}.
$$