$($theta$),$ as shown below in Figure $1.$

The force equation for pendulum is

mgsin$+mL\frac{d^2}{d{t}^2}=0$

Thus $\frac{d^2}{d{t}^2}=-\frac{g}{L}sin.$

For all angles $,$ we make the assumption that $sin=$ leading to an analytical solution. However, for large angles $,$ we resort to a numerical method to solve the differential equation above. We first reduce the second$-$order differential equation to a set of two first$-$order differential equations by introducing $($angular velocity$),$ thus