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.)