The initial value problem for the damped pendulum equation is 2 where l is the length of the pendulum, k (with [k] = T'l) is the air resistance (or friction constant) and g is the gravitational constant. The initial position of the pendulum is given by the angle 6'0 and its initial velocity is given by we. (lo) By neglecting the kg term and setting sin(l9) = 6', nd the characteristic time scale appropri- ate to a small-amplitude oscillatory behaviour. Using this characteristic time scale and taking le = 90 write down the rescaled problem together with the new initial conditions