The two ends of a uniform rod are insulted and the temperature is initially 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.

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

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

n=f(x)-xm

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

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