Regarding steady state solution of $u_{t}= c^2 u_{xx}$ with $u_{x}(0,t) = c_{1}$ and $u(L,t) = c_{1}$? Suppose we have the one dimensional diffusion equation $u_{t} = c^2 u_{xx}$ with the boundary condition $u(L,t) = c_{2} $ and $u_{x}(0,t) = c_{1}$. I donot recognize which type of condition is it? badly, it seems to be the Robin condition.
Suppose I proceed by assuming $u(x,t) = X(x) T(t)$ then $XT' = c^2 X'' T$, but I am thinking how to incorporate the conditions given into the PDE or the process which can give us the steady state solution?
EDIT:
I am still thinking of the solution, the question is attached as pic.

The solution $u(x,t)$ is independent of $t$ in the below solution, may be the steady state then also how to answer the 2nd part as it hints the dependence of time there:


Let $u(x,t) = w(x) + v(x,t)$ where $w(x)$ is the steady-state. Then we have
$$ w''(x) = 0, \quad w'(0) = w(L) = c_1 $$
which gives $w(x) = c_1(x-L) + c_1$
Now you can use separation of variables to find $v(x,t)$, which is homogeneous on the boundary (of "mixed" type).
