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

The Correct Option is A

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