The two ends of a uniform rod are insulted and the temperature is initially $\displaystyle u(x)=f(x)$. Find the equilibrium temperature which the rod assumes after a long time.

Here is what I have which isn't work so is more than likely wrong.

$\displaystyle \text{B.C.}: \ u(x)=f(x)$

$\displaystyle \displaystyle\text{D.E.}: \ \frac{d^2u}{dx^2}=0\Rightarrow u(x)=xm+n$

$\displaystyle n=f(x)-xm$

$\displaystyle u(x)=xm+f(x)-xm\Rightarrow u(x)=f(x)$

$\displaystyle \displaystyle\lim_{x\to\infty}u(x)=f(\infty)$