The second question in my book and I've not the foggiest idea where to begin:
Suppose the rod has a constant internal heat source, so that the basic equation describing the heat flow within the rod is
$$u_t = \alpha ^2 u_{xx} + 1 \qquad for \qquad 0 \lt x \lt 1$$
Suppose we fix the boundaries' temperatures by $u(0, t) = 0$ and $u(1, t) = 1$. What is the steady-state temperature of the rod? In other words, does the temperature $u(x, t)$ converge to a constant temperature $U(x)$ independent of time?
What I know: It seems that if our equation $u(x, t)$ satisfies the boundary conditions, then we can assume that it is not dependent on time. For example, a structure of $u(x, t)$ satisfying the boundary conditions would be something of the form $u(x, t) = x$, so naturally $u_t = 0$. However, I've not any idea how this aids me in solving the problem, and I am not necessarily sure how to graph the PDE. Any help here? They got some wonky answer: $U(x) = - \frac{1}{2\alpha^2} (x^2 - x) +x$.