Using MATLAB, simulate the response of both the linear and the nonlinear differential equation

models for the simple pendulum discussed in class. Run the following cases. Use $L=0\mathrm{.}5m,$ and $g$

$=9\mathrm{.}81\frac{m}{s^2}.$

a$.{}_0\frac{?}{P}=$I $\frac{?}{8\text{r}\text{a}\text{d}}$ and ${}_0^{}=0ra\frac{d}{s}$

b$.{}_0\frac{?}{P}=$I $\frac{?}{4\text{r}\text{a}\text{d}}$ and ${}_0^{}=0ra\frac{d}{s}$

For each case, provide a properly labeled plot of the angular position $($rad$)$ and angular velocity

$(ra\frac{d}{s})$ on the same graph over the interval of $0$ to $3sec.$ Compare the response $($i$.$e$.,$ motion

range, and frequency$)$ between the linear and nonlinear models for both cases. Note that the

model for the pendulum is a nonlinear differential equation. Below is an example of how to

solve a non$-$linear differential equation in MATLAB. The example is:

$y^{}-10cosy=0$

Modify this code to represent the equation for the pendulum.

$$ Solving a nonlinear differential equation: ydotdot $-10cosy=0$

function example

tspan$=[0,5]$;$,$ Define the time interval

$y0=[0$;$0]$;$,\frac{\text{@}}{0}$ Define the initial conditions for $y$ and $y$ dot

$[t,y]=$ ode$45($@nf$,$ tspan, y$0)$; Call function ode$45$ to solve equation

P Plot of the solution

plot $(t,y($:$,1),{?}^{\text{'}}{k}^{-{?}^{\text{'}}},t,y($:$,2),{?}^{\text{'}}k--{?}^{\text{'}})$; $\%$ plot $y$ and $y_{d}ot$

$\%$ Add label

$$ Add legend

:$.$ Add title

$\%$ Add your name to each plot

end $\%$end of example

$--------$

function dydt $=nf(t,y)$

dydt $=[y(2)$;$10\mathrm{.}0**cos(y(1))]$;$,$ define the differential equation here

end $\%$end of dydt

For modeling the linear motion of the pendulum, create another copy of the nf

function for the pendulum calling it lf$,$ and replace sin$($y$(1))$ with just y$(1).$