Step 1: A 1-metre copper rod is in steady state. Its left end is held at 100°C and its right end at 0°C. Find the temperature at x = 0.5 m. Step 2: Steady state means nothing changes with time, so ∂u/∂t = 0. The heat equation then reduces to Laplace's equation in one dimension: d²T/dx² = 0. Step 3: Integrate twice: dT/dx = A, so T(x) = Ax + B. Step 4: Apply boundary conditions. At x = 0, T = 100°C, so B = 100. At x = 1, T = 0°C, so A + 100 = 0, giving A = −100. Step 5: The temperature profile is T(x) = 100 − 100x. Step 6: At x = 0.5 m: T = 100 − 100(0.5) = 50°C. Result: The midpoint of the rod is at 50°C, exactly halfway between the two ends, as expected from the linear profile.