PDEs, separation of variables, and the heat equation

Find a series solution of

Find a series solution of

Use separation of variables to find a nontrivial solution to $u_{xt} = u_{xx}$.

Use a variation on separation of variables to find a nontrivial solution to $u_{x} + u_{t} = u$. Hint: Try $u(x,t) = X(x)+T(t)$.

Suppose that the temperature on the wire is fixed at $0$ at the ends, $L=1$, $k=1$, and $u(x,0) = 100\sin(2 \pi x)$.

1. What is the temperature at $x = \nicefrac{1}{2}$ at any time?
2. What is the maximum and the minimum temperature on the wire at $t=0$?
3. At what time is the maximum temperature on the wire exactly one half of the initial maximum at $t=0$?