In my PDE book, in order to solve nonhomogeneous pde's we start off (with a nonhomogeneous heat equation) by obtaining an equilibrium temperature $u_E(x)$. It's time independent, and must satisfy
$$\frac{d^2u_E}{dx^2} = 0 \tag{A}$$
Which is something I've seen before, but I've never really thought about the meaning of (A). I get that it means that the system has reached a state such that the temperature gradient is stable in time. But I'm having trouble reconciling that with the math. For example in a 1D rod, couldn't I have a $u_E(x)$ of the form $u_E = x^3$? If so, $u_E$ is definitely not going to satisfy (A). Are all steady state temperature distributions linear?