My problem is given as
Arbitrary temperatures at ends . If the ends $x=0$ and $x=L$ of the bar in the text are kept at constant temperatures $U_1$ and $U_2$ respectively, what is the temperature $u_1(x)$ in the bar after a long time (theoretically, as $t \to \infty$)? First guess, then calculate.
My guess is that the temperature after a very long time is given as the meadian temperature. Etc
$u(x,t) \approx (U_2-U_1)/L, \quad \text{as} \quad t\to \infty$
Now one does assume that the temperature reaches a limit, which is not unlikely, then the solution will satisfy the laplacian $\nabla^2u=0$.
Which leads to the heat equation in one variable
$ \frac{\mathrm{d}u}{\mathrm{d}t} = c^2 \frac{\mathrm{d}^2u}{\mathrm{d}x^2} $
The standard way of assuming the solution is on the form $u(x,t)=X(x)T(t)$ fails for me.Begin with assuimg that the differential equation is equal to some arbitary constant $\lambda$ that is not dependant on $x$ nor $t$. Then I end up with the set of equations
$\begin{array}{lcr} T' & = & \lambda c^2 T\\ \ddot{X} & = & \lambda X \end{array}$ If we assume for a minute that $\lambda=0$, we end up with $X(x) = Ax + B, \qquad T(t)=C$ Which does not satifsy the initial values. So, what do I do to solve this bugger?