### A Level Further Maths: Simple Harmonic Motion - SHM: Light Damping

The following example shows how to model light damping by using a second order differential equation. The second-order differential equation modelling light damping is given by: \[ \ddot{x} + 0.5 \dot{x} + 4 x = 0 \] The characteristic equation is: \[ m^2 + 0.5m + 4 = 0 \] Solving for \(m\), we get: \[ m = -0.25 \pm i \sqrt{3.9375} \] Therefore, the general solution to the differential equation is: \[ x(t) = e^{-0.25t} \left( \cos(\sqrt{3.9375} \, t) + 0.1259 \sin(\sqrt{3.9375} \, t) \right) \] where \( A = 1 \) and \( B \approx 0.1259 \) are determined by the initial conditions: Initial displacement \( x(0) = 1 \) m Initial velocity \( \dot{x}(0) = 0 \) m/s The above equation produces a curve that shows light damping in simple harmoinc motion - SHM. If you want to play with it interactively, plea