Classical Dynamics of Particles and Systems · Problem 3-36

The key is recognizing that any solution to the harmonic oscillator equation can be written as a linear combination of two independent solutions—you're just shifting the initial time from $t=0$ to $t=t_0$ and adjusting the constants accordingly. Start by writing the general solution in terms of $\sin$ and $\cos$ functions (or complex exponentials), then apply your two initial conditions at $t=t_0$ to solve for the amplitude and phase (or the two coefficients). This gives you the displacement $x(t)$ valid for all $t \geq t_0$, expressed entirely in terms of $x_0$, $\dot{x}_0$, and the oscillator's natural frequency.

The key is recognizing that any solution to the harmonic oscillator equation can be written as a linear combination of two independent solutions—you're just shifting the initial time from $t=0$ to $t=t_0$ and adjusting the constants accordingly. Start by writing the general solution in terms of $\sin$ and $\cos$ functions (or complex exponentials), then apply your two initial conditions at $t=t_0$ to solve for the amplitude and phase (or the two coefficients). This gives you the displacement $x(t)$ valid for all $t \geq t_0$, expressed entirely in terms of $x_0$, $\dot{x}_0$, and the oscillator's natural frequency.