Question

Let $c$ be a positive real number and let $u: \mathbb{R}^2 \to \mathbb{R}$ be defined by $u(x,t) = \frac{1}{2c} \int_{x-ct}^{x+ct} e^{s^2} \, ds \quad \text{for} \ (x,t) \in \mathbb{R}^2.$ Then which one of the following is true?

• A. $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \ \text{on} \ \mathbb{R}^2.$
• B. $\frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2} \ \text{on} \ \mathbb{R}^2.$
• C. $\frac{\partial u}{\partial t} \frac{\partial u}{\partial x} = 0 \ \text{on} \ \mathbb{R}^2.$
• D. $\frac{\partial^2 u}{\partial t \partial x} = 0 \ \text{on} \ \mathbb{R}^2.$

Answer 1

Answer: (A)

$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \ \text{on} \ \mathbb{R}^2.$

Answer 2

The correct option is (A): $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \ \text{on} \ \mathbb{R}^2.$