# Extra problems for Math 421

Here are some problems used by Professor Wheeden for Section
13.5 *Laplace's Equation*.

(**A**) Solve the eigenfunction problem
$$
X''+\lambda X=0
$$
for 0 < x < L with X(0) =
X'(L) = 0. Consider all possible cases.

(**B**) Find the steady state temperature for the heat
problem
$$
u_t = 4 u_{xx},\quad 0 < x < 10,\quad t > 0,
$$
with the boundary conditions u(0,t) = 200 and
u(10,t)= 300 for t >
0, and with initial condition u(x,0)
=f(x) for 0 < x <
10. (*Hint*: subtract an appropriate linear function
of x from u(x,t) to reduce to the case of
zero boundary condition.)