Question

Consider two particles moving along the x-axis. In terms of their coordinates \( x_1 \) and \( x_2 \), their velocities are given as \( \frac{dx_1}{dt} = x_2 - x_1 \) and \( \frac{dx_2}{dt} = x_1 - x_2 \), respectively. When they start moving from their initial locations of \( x_1(0) = 1 \) and \( x_2(0) = -1 \), the time dependence of both \( x_1 \) and \( x_2 \) contains a term of the form \( e^{at} \), where \( a \) is a constant. The value of \( a \) (an integer) is:

Hint:

For coupled differential equations, look for solutions that involve exponential terms, and solve for constants based on initial conditions.

Answer 1

Correct Answer: -2

Step 1: Analyzing the system.
We are given a system of two particles with velocities dependent on the relative positions of the particles. The differential equations are: \[ \frac{dx_1}{dt} = x_2 - x_1 \quad \text{and} \quad \frac{dx_2}{dt} = x_1 - x_2 \] This is a coupled system that can be solved by linear algebra methods. Step 2: Solving the differential equations.
By solving these differential equations, the time dependence of both \( x_1(t) \) and \( x_2(t) \) can be found to be of the form \( e^{at} \), with \( a = -2 \).