1 too diffuse
Let's revisit heat flow in a metal bar. With uniform heating $s$ throughout the bar, we have
$\frac{\partial T}{\partial t} = \kappa \frac{\partial^2 T}{\partial x^2} + s$ with $T = 0$ at $x = 0, L$ and $T = 1$ at $t = 0$.
a. Start by finding the steady state solution.
b. Then, looking for a perturbation away from the steady solution, use separation of variables to
find the ODEs in $x$ and $t$.
c. determine the eigenvalues and compare them to the eigenvalues we derived in class.
d. write out the full solution for the temperature including the undetermined coefficients.
e. use orthogonality to determine the coefficients using the initial condition.
