Solving non-homogeneous PDE (unsure of methodology)

[King Tony]

Homework Statement

$$u_{t} = ku_{xx}$$
$$u_{x}(0, t) = 0$$
$$u_{x}(L, t) = B =/= 0$$
$$u(x, 0) = f(x)$$

Homework Equations

The Attempt at a Solution

I believe that no equilibrium solution exists because we can't solve
$$u_{xx} = 0$$
with our boundary conditions. I'm a little lost as to where to take this question from here.

Been trying to work with this question for around 30 minutes now, I'm lost. :D

 

[King Tony]

Think I figured it out now, derp...

I just chose a reference function
$$r(x, t) = r(x) = c_{1}\frac{x^2}{2}$$
and solved for c1 which allowed me to generate a new linear homogeneous PDE and summed up the solution and the reference function to find the final solution. Would be nice if someone replied that I used the correct method though! Thanks!

 

[HallsofIvy]

Yes. Find a function that satisfies the boundary conditions with regard to the differential equation, then subtract it off to get a new differential equation with homogeneous boundary conditions.