Problem I'm Facing: If it exists, find the steady-state temperature solution for the PDE $$u_t = u_{xx} + 1$$ with the boundary conditions $$u_x (0,t) = 1 \;\;\;\;\; u(L,t) = 1-L$$
This has come up in a homework assignment in my class on PDEs. However, the professor isn't teaching much of the underlying theory or methods and is making the lectures extremely hard to follow, so this is ultimately feeling more like a self-study than anything, so I'm not feeling very confident.
So I have tried to attempt the above problem. As given in our text, a steady-state solution is one that will not vary with time. Let $U(x)$ be such a solution: then, per the text, we must have
$$kU''(x) = 0 \implies U(x) = c_1 x + c_2$$
for constants $k,c_1,c_2$, and of course the time derivative is zero.
In that light, letting $u(x,t) = U(x)$ and using the above, we seem to get
$$u_t = u_{xx} + 1 \implies 0 = 0 + 1 \implies 0 = 1$$
Obviously this is not true, so this seems to imply no such steady state solution exists.
But my lack of confidence makes me feel like that this is just way too easy and that I'm completely going the wrong way with this. Is this at all right? How exactly would I find the solution?