Question

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

• A. \( \frac{\partial^2 u}{\partial t^2} = c^2 \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) + xy \)
• B. \( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0 \)
• C. \( \frac{\partial u}{\partial t} = c \frac{\partial u}{\partial x} \)
• D. \( \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \)

Hint:

To classify a PDE, check the order of the highest derivative, linearity, and whether the equation is homogeneous.

Answer 1

The Correct Option is B

A second-order, linear, homogeneous partial differential equation has the following characteristics:

• The highest derivatives are of the second order.
• The equation is linear, meaning it has no products or nonlinear functions of the dependent variable or its derivatives.
• The equation is homogeneous, meaning all terms involve the dependent variable or its derivatives.

Now, let's analyze the given equations:

• (A) This equation has a non-homogeneous term \( xy \), so it is not homogeneous.
• (B) This is a second-order, linear, homogeneous partial differential equation, as it involves second derivatives and has no non-homogeneous term.
• (C) This is a first-order equation, so it does not meet the criteria for a second-order equation.
• (D) This equation is second-order, linear, and homogeneous, but it is simpler and only involves spatial derivatives, not the time derivative. However, it still qualifies as a second-order PDE.

Thus, the correct answer is option (B).

Equations like (B) represent the Laplace equation, which is commonly encountered in fields such as physics and engineering.