Chapter 5.2, Problem 73ES

Suppose that a uniform metal rod $50\text{ cm}$ long is insulated laterally, and the temperatures at the exposed ends are maintained at $25\text{°C}$ and $85°\text{C}$ , respectively. Assume that an $x\text{-axis}$ is chosen as in the accompanying figure and that the temperature $T\left(x\right)$ satisfies the equation

$\frac{d^{2}T}{d{x}^2}=0$

Find $T\left(x\right)$ for $0\le x\le 50$ .