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Which of the following equations belong/belongs to the class of second-order, linear, homogeneous partial differential equations:

1. ∂²u/∂t² = c²(∂²u/∂x² + ∂²u/∂y²) + xy
2. ∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z² = 0
3. ∂u/∂t = c ∂u/∂x
4. (∂²u/∂t²)² = c² ∂²u/∂x²

Correct answer: ∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z² = 0

Solution

The correct option is a second-order, linear, homogeneous partial differential equation because it consists of second derivatives of the function u, is linear in u and its derivatives, and equals zero, indicating that there are no additional terms that would make it non-homogeneous.

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