Consider two simple harmonic oscillators with the same spring constant k and mass m. (Damping and driving forces are absent.) One oscillator is started with initial conditions
x1(0) = x0
dx1/dt (0) = v0;
the other starts with slightly different conditions
x2(0) = x0 + n
dx2/dt (0) = v0 + h
where n and h are arbitrary small changes in the initial position and velocity, respectively.
a. Find the difference in the oscillators' positions, x2(t) - x1(t), for all t >= 0.
b. This difference is bounded, i.e., there exists a constant C, independent of time, for which |x2(t) - x1(t)| <= C holds for all time. Find an expression for C. (Most credit will be awarded for the best bound, i.e., the smallest value of C which works.)