Problem #4 (20 points) - A Heat-Like Equation
Solve for $u(x, t)$ in the region
$R = \{(x,t)|0 \le x \le 1,0 \le t\}$
given the partial differential equation,
$x^2 \frac{\partial^2 u(x,t)}{\partial x^2} + x \frac{\partial u(x,t)}{\partial x} - 4u(x,t) = x^2 \frac{\partial u(x,t)}{\partial t}$,
the boundary conditions:
$u(0,t) = \text{finite}$ , $u(1,t) = 1$
and the initial condition,
$u(x, 0) = x^2 + 1$.
Note that you need not evaluate the integrals in the dot product in your final solution.
