Question

For the differential equation given below, which one of the following options is correct? \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \quad 0 \leq x \leq 1, 0 \leq y \leq 1 \]

• A. \( u = e^{x + y} \) is a solution for all \(x\) and \(y\)
• B. \( u = e^{x} \sin y \) is a solution for all \(x\) and \(y\)
• C. \( u = \sin x \sin y \) is a solution for all \(x\) and \(y\)
• D. \( u = \cos x \cos y \) is a solution for all \(x\) and \(y\)

Hint:

The Laplace equation is a partial differential equation that requires the sum of the second derivatives with respect to each variable to be zero. Common solutions include trigonometric functions such as sine and cosine.

Answer 1

The Correct Option is C

We are given the Laplace equation in two variables \(x\) and \(y\): \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \] We need to check which function satisfies this equation. 1. Option (A): \( u = e^{x + y} \)
First, we compute the second derivatives: \[ \frac{\partial^2 u}{\partial x^2} = e^{x + y}, \quad \frac{\partial^2 u}{\partial y^2} = e^{x + y} \] Therefore, the sum of the second derivatives is: \[ e^{x + y} + e^{x + y} = 2e^{x + y} \neq 0 \] So, option (A) does not satisfy the Laplace equation. 2. Option (B): \( u = e^{x} \sin y \)
We compute the second derivatives: \[ \frac{\partial^2 u}{\partial x^2} = e^x \sin y, \quad \frac{\partial^2 u}{\partial y^2} = -e^x \sin y \] Therefore, the sum of the second derivatives is: \[ e^x \sin y - e^x \sin y = 0 \] So, option (B) satisfies the Laplace equation. 3. Option (C): \( u = \sin x \sin y \)
We compute the second derivatives: \[ \frac{\partial^2 u}{\partial x^2} = -\sin x \sin y, \quad \frac{\partial^2 u}{\partial y^2} = -\sin x \sin y \] Therefore, the sum of the second derivatives is: \[ -\sin x \sin y - \sin x \sin y = 0 \] So, option (C) satisfies the Laplace equation. 4. Option (D): \( u = \cos x \cos y \)
We compute the second derivatives: \[ \frac{\partial^2 u}{\partial x^2} = -\cos x \cos y, \quad \frac{\partial^2 u}{\partial y^2} = -\cos x \cos y \] Therefore, the sum of the second derivatives is: \[ -\cos x \cos y - \cos x \cos y = -2 \cos x \cos y \neq 0 \] So, option (D) does not satisfy the Laplace equation. Thus, the correct answer is (C).