In Problems 5-8, find the steady-state temperature $u(r, z)$ in a
finite cylinder defined by $0 \le r \le 1, 0 \le z \le 1$ if the boundary
conditions are as given.
5. $u(1, z) = z$, $0 < z < 1$
6. $u(1, z) = z$, $0 < z < 1$
$\frac{\partial u}{\partial z}|_{z=0} = 0$, $0 < r < 1$
$\frac{\partial u}{\partial z}|_{z=0} = 0$, $0 < r < 1$
$\frac{\partial u}{\partial z}|_{z=1} = 0$, $0 < r < 1$
$u(r, 1) = 0$, $0 < r < 1$
7. $u(1, z) = u_0$, $0 < z < 1$
8. $u(1, z) = 0$, $0 < z < 1$
$u(r, 0) = 0$, $0 < r < 1$
$\frac{\partial u}{\partial z}|_{z=0} = 0$, $0 < r < 1$
$\frac{\partial u}{\partial z}|_{z=1} = 0$, $0 < r < 1$
$u(r, 1) = u_0$, $0 < r < 1$
