The differential equation for the angle as a function of time is
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It is not possible to write a formula for the solution to this equation in terms of an elementary function. Instead, we use an approximation which is fairly accurate if the angle is sufficiently small (i.e. when ):
.
Solving this differential equation allows us to find formula for the angle of the pendulum at a given time :
and the angular speed at a given time :
where is the initial angle of the pendulum. Note that the angular frequency of the pendulum is a constant.