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:

When solving coupled linear differential equations, the solutions often contain exponential terms of the form \( e^{at} \), where \( a \) is determined by the eigenvalue of the system.

Answer 1

Step 1: Set up the differential equations.
We are given the following system of differential equations: \[ \frac{dx_1}{dt} = x_2 - x_1, \quad \frac{dx_2}{dt} = x_1 - x_2 \] These equations are coupled and linear. We can solve them by taking the second derivative of both equations and eliminating \( x_1 \) and \( x_2 \).
Step 2: Solve the system.
Taking the derivative of \( \frac{dx_1}{dt} \) with respect to \( t \) gives: \[ \frac{d^2x_1}{dt^2} = \frac{dx_2}{dt} - \frac{dx_1}{dt} = (x_1 - x_2) - (x_2 - x_1) \] This simplifies to: \[ \frac{d^2x_1}{dt^2} = 2(x_1 - x_2) \] Now, solving this system gives a solution of the form \( x_1(t) = A e^{at} + B \), where \( a \) is the constant we need to determine.
Step 3: Conclusion.
The value of \( a \) is 1.