Solve Nonhomogenous PDE: Equilibrium Temp Distribution, B Value

[physstudent.4]

Homework Statement

du/dt=(d^2 u)/dx^2+1
u(x,0)=f(x)
du/dx (0,t)=1
du/dx (L,t)=B
du/dt=0
Determine an equilibrium temperature distribution. For what value of B is there a solution?

Homework Equations

Not really sure what to put here.

The Attempt at a Solution

I started by trying to separate variables, with u(x,t)=phi(x)*g(t), and got to
g'(t)/g(t)=phi''(x)/phi(x)+1/(phi(x)g(t))=0.
So g(t) is constant based on the above, but then I get a little lost while trying to solve for phi. I tried letting g(t)=lamda (abbreviated lm from now on), and got

phi''(t)+1/lm=0, which yields a quadratic solution of [(-lm*x^2)/2+x/lm+C/lm] after using the condition du/dx (0,t)=1. Then, since this is phi(x),
u(x,t)= -lm^2*x^2/2+ x + lm*C.
Using the other condition du/dx (L,t)=B, and assuming some quick mental algebra was correct,
B=-lm^2*L+1.
First off, is all of the above a correct approach as far as you can tell? And secondly, do I need to find u(x,t) or a specific value of lm?

 

[HallsofIvy]

I am confused by the question. You say "Determine an equilibrium temperature distribution. For what value of B is there a solution?"

It is very easy to determine the general equilibrium solution. Is the second independent of that or is it asking for a value of B for which there is an equilibrium solution?

 

[physstudent.4]

I found the equilibrium solution easily enough after posting that, I realized I could solve for lm. Another student and I clarified with our professor today, he wants both, which I now have. Thanks though!