Question

Identify the following partial differential equations in the given order based on their generic form where \( c \) is a constant, and \( u_1 \) and \( u_2 \) are functions of \( (x, t) \) and \( (x, y) \), respectively:
(i) \[ \frac{\partial^2 u_1}{\partial t^2} = c^2 \frac{\partial^2 u_1}{\partial x^2} \] (ii) \[ \frac{\partial u_1}{\partial t} = c^2 \frac{\partial^2 u_1}{\partial x^2} \] (iii) \[ \frac{\partial^2 u_2}{\partial x^2} + \frac{\partial^2 u_2}{\partial y^2} = 0 \]

• A. (i) Wave equation, (ii) Heat equation, (iii) Laplace equation
• B. (i) Heat equation, (ii) Laplace equation, (iii) Wave equation
• C. (i) Laplace equation, (ii) Wave equation, (iii) Heat equation
• D. (i) Heat equation, (ii) Wave equation, (iii) Laplace equation

Hint:

Match each equation to its standard form: second time derivative for wave, first time derivative for heat, and pure spatial Laplacian equal to zero for Laplace.

Answer 1

The Correct Option is A

This question checks if you can recognize the three classic second-order PDEs from their generic forms.

1. Equation (i): \( u_{1tt} = c^2 u_{1xx} \) has a second derivative in time equal to a constant times the second derivative in space. This is the standard wave equation, which models vibrations and traveling waves.
2. Equation (ii): \( u_{1t} = c^2 u_{1xx} \) has only a first derivative in time balanced against a second derivative in space. This is the heat, or diffusion, equation, which models temperature or concentration spreading over time.
3. Equation (iii): \( u_{2xx} + u_{2yy} = 0 \) has no time variable at all, just the sum of two pure second spatial derivatives set to zero. This is the Laplace equation, which describes steady-state fields.

So the order is wave, heat, Laplace, which matches option (A).