The pendulum oscillation is described by the initial$-$value problem consisting of the equation of motion

with initial conditions:

${}^{}+\frac{g}{L}sin()=0,(t_0)={}_0,{}^{}(t_0)={}_0$

Here, $L$ is the length of the pendulum, $g$ acceleration due to gravity, $$ the angle the pendulum makes

with the vertical, ${}_0$ the initial angular displacement, and ${}_0$ the initial angular velocity.

a$)$ Transform the second$-$order initial$-$value problem into a system of first$-$order initial$-$value

problems.

b$)$ Compute the angular displacement $(0\mathrm{.}2)$ and angular velocity ${}^{}(0\mathrm{.}2)$ using the forward

Euler method with step size $h=0\mathrm{.}1.$ Use $L=0\mathrm{.}5m,g=9\mathrm{.}81\frac{m}{s^2},t_0=0,{}_0=\frac{}{4}$rad, and ${}_0$

$=0ra\frac{d}{s}.$help please A simple pendulum is idealized by considering a bob mass swinging on an inextensible, massless rod $($negligible mass relative to the bob$)$ about a frictionless pivot $($Figure $1).$ Figure $1$ Simple pendulum The pendulum oscillation is described by the initial$-$value problem consisting of the equation of motion with initial conditions:

$\backslash$theta $+$g$/$Lsin $(\backslash$theta $)=0,\backslash$theta $($t$\_0)=\backslash$theta $\_0,\backslash$theta $($t$\_0)=\backslash$omega $\_0$

Here, L is the length of the pendulum, g acceleration due to gravity, $\backslash$theta the angle the pendulum makes with the vertical, $\backslash$theta $\_0$ the initial angular displacement, and $\backslash$omega $\_0$ the initial angular velocity. a$)$ Transform the second$-$order initial$-$value problem into a system of first$-$order initial$-$value problems. b$)$ Compute the angular displacement $\backslash$theta $(0\mathrm{.}2)$ and angular velocity $\backslash$theta $(0\mathrm{.}2)$ using the forward Euler method with step size h$=0\mathrm{.}1.$ Use L$=0\mathrm{.}5$m$,$ g$=9\mathrm{.}81$m $/$ s$^2,$ t$\_0=0,\backslash$theta $\_0=\backslash$pi $/4$ r a d$,$ and $\backslash$omega $\_0=0$ rad $/$ s$.$