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 problem involves evaluating the function \( u(x, t) \) and determining which option correctly represents a relationship between its partial derivatives. Let's analyze the given function:

The function \( u(x, t) \) is defined by:

\(u(x,t) = \frac{1}{2c} \int_{x-ct}^{x+ct} e^{s^2} \, ds\)

We need to determine which of the given options correctly describes a differential equation or property involving the function's derivatives. Noticing that the function involves integration with respect to \( s \), it hints that \( u(x, t) \) might satisfy a type of wave equation which is commonly expressed as:

\(\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}\)

Let's check this claim by evaluating the partial derivatives of \( u(x, t) \).

1. First Partial Derivatives:

• Start by calculating \( \frac{\partial u}{\partial t} \): 
  Using Leibniz's Rule for differentiating under the integral, \(\frac{\partial u}{\partial t} = \frac{1}{2c} \left( -ce^{(x-ct)^2} + ce^{(x+ct)^2} \right) = \frac{1}{2}(e^{(x+ct)^2} - e^{(x-ct)^2})\)
• Calculate \( \frac{\partial u}{\partial x} \): 
  Using Leibniz's Rule again, \(\frac{\partial u}{\partial x} = \frac{1}{2c} \left( e^{(x+ct)^2} + e^{(x-ct)^2} \right)\)

2. Second Partial Derivatives:

• Calculate \( \frac{\partial^2 u}{\partial t^2} \): 
  Differentiate \( \frac{\partial u}{\partial t} \) with respect to \( t \), applying the chain rule and product rule as needed: \(\frac{\partial^2 u}{\partial t^2} = c(e^{(x+ct)^2} + e^{(x-ct)^2})\)
• Calculate \( \frac{\partial^2 u}{\partial x^2} \): 
  Differentiate \( \frac{\partial u}{\partial x} \) with respect to \( x \): \(\frac{\partial^2 u}{\partial x^2} = \frac{1}{c} \left( (x+ct)e^{(x+ct)^2} + (x-ct)e^{(x-ct)^2} \right)\)

Comparing:

We observe that:

• \(\frac{\partial^2 u}{\partial t^2}\) simplifies to a form resembling the wave equation: 
  \(\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}\),

Thus, this differential equation confirms that \( u(x, t) \) satisfies the wave equation as defined in the first option. Consequently, the correct answer is:

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

Hence, option (A) is the correct choice.