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The solution of the partial differential equation ∂u/∂t = α ∂²u/∂x² is of the form

1. C cos(kt) [C1 e^(√(k/α) x) + C2 e^(-√(k/α) x)]
2. C e^(kt) [C1 e^(√(k/α) x) + C2 e^(-√(k/α) x)]
3. C e^(kt) [C1 cos(√(k/α) x) + C2 sin(-√(k/α) x)]
4. C sin(kt) [C1 cos(√(k/α) x) + C2 sin(-√(k/α) x)]

Correct answer: C e^(kt) [C1 e^(√(k/α) x) + C2 e^(-√(k/α) x)]

Solution

The correct option represents a solution to the heat equation, which describes how heat diffuses through a medium. The exponential term e^(kt) indicates growth over time, while the combination of e^(√(k/α) x) and e^(-√(k/α) x) captures the spatial behavior of the solution, consistent with the characteristics of the equation.

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