Suppose that a uniform metal rod 50 cm long is insulated laterally, and the temperatures at the

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Suppose that a uniform metal rod 50 cm long is insulated...

Suppose that a uniform metal rod $50 \mathrm{cm}$ long is insulated laterally, and the temperatures at the exposed ends are maintained at $25^{\circ} \mathrm{C}$ and $85^{\circ} \mathrm{C}$, respectively. Assume that an $x-$ axis is chosen as in the accompanying figure and that the temperature $T(x)$ satisfies the equation $$\frac{d^{2} T}{d x^{2}}=0$$ Find $T(x)$ for $0 \leq x \leq 50$

Suppose that a uniform metal rod $50 \mathrm{cm}$ long is insulated laterally, and the temperatures at the exposed ends are maintained at $25^{\circ} \mathrm{C}$ and $85^{\circ} \mathrm{C}$, respectively. Assume that an $x-$ axis is chosen as in the accompanying figure and that the temperature $T(x)$ satisfies the equation
$$\frac{d^{2} T}{d x^{2}}=0$$
Find $T(x)$ for $0 \leq x \leq 50$

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Key Concepts

Steady-State Heat Conduction

In problems involving heat conduction, a steady state implies that the temperature distribution within the medium does not change with time. This leads to the elimination of time-dependent terms, resulting in a spatial differential equation for temperature that reflects the equilibrium condition throughout the material.

Second Order Differential Equations and Linear Solutions

The differential equation d²T/dx² = 0 is a second order linear ordinary differential equation with constant coefficients. The general solution to this equation is a linear function, T(x) = Ax + B, demonstrating that when the rate of heat transfer is steady and the material is uniform, the temperature varies linearly with position.

Boundary Conditions

Boundary conditions are necessary to uniquely determine the constants in the general solution of a differential equation. In the context of temperature distribution, specifying the temperatures at the ends of a rod (Dirichlet boundary conditions) allows for the complete determination of the linear temperature profile within the rod.

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