Question

Consider the following first-order partial differential equation, also known as the transport equation \[ \frac{\partial y(x,t)}{\partial t} + 5 \frac{\partial y(x,t)}{\partial x} = 0 \] with initial conditions given by \( y(x, 0) = \sin x, -\infty < x < \infty \). The value of \( y(x,t) \) at \( x = \pi \) and \( t = \frac{\pi}{6} \) is

• A. 1
• B. 2
• C. 0
• D. 0.5

Hint:

In transport equations, the solution moves along the characteristic lines, and the initial condition defines the shape of the solution.

Answer 1

The Correct Option is D

The given equation is a transport equation, which describes the evolution of a function \( y(x,t) \) as it moves along the \( x \)-axis. The general solution to this equation can be written as: \[ y(x,t) = f(x - 5t), \] where \( f(x) \) is the initial condition function at \( t = 0 \). From the given initial condition \( y(x, 0) = \sin x \), we know that \( f(x) = \sin x \). Therefore, the solution to the equation is: \[ y(x,t) = \sin(x - 5t). \] Now, we evaluate \( y(x,t) \) at \( x = \pi \) and \( t = \frac{\pi}{6} \): \[ y(\pi, \frac{\pi}{6}) = \sin \left( \pi - 5 \times \frac{\pi}{6} \right) = \sin \left( \pi - \frac{5\pi}{6} \right) = \sin \left( \frac{\pi}{6} \right). \] Since \( \sin \left( \frac{\pi}{6} \right) = \frac{1}{2} \), the value of \( y(x,t) \) at \( x = \pi \) and \( t = \frac{\pi}{6} \) is \( 0.5 \).
Thus, the correct answer is Option (D): \( 0.5 \).