Question

The function \( u(x, t) \) satisfies the initial value problem \[ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2}, \, x \in \mathbb{R}, \, t > 0, \] \[ u(x, 0) = 0, \, \frac{\partial u}{\partial t} (x, 0) = 4xe^{-x^2}. \] Then \( u(5, 5) \) is:

• A. \( 1 - \frac{1}{e^{100}} \)
• B. \( 1 - e^{100} \)
• C. \( 1 - \frac{1}{e^{10}} \)
• D. \( 1 - e^{10} \)

Hint:

For wave equations, use the method of characteristics to solve for \( u(x, t) \) and apply initial conditions to find specific solutions.

Answer 1

The Correct Option is A

We are given the wave equation, and we are asked to solve for \( u(5, 5) \) using the initial conditions. The general solution to the wave equation is of the form: \[ u(x, t) = f(x - t) + g(x + t), \] where \( f \) and \( g \) are determined from the initial conditions. Applying the initial conditions: - \( u(x, 0) = 0 \) gives \( f(x) + g(x) = 0 \), so \( g(x) = -f(x) \). - \( \frac{\partial u}{\partial t}(x, 0) = 4xe^{-x^2} \) gives \( -f'(x) + g'(x) = 4xe^{-x^2} \). Solving these equations, we find: \[ u(x, t) = 1 - \frac{1}{e^{100}}. \] Final Answer: (A) \( 1 - \frac{1}{e^{100}} \).