$($a$)$ Modify the function ex$\_$with $2$eqs to solve the IVP $(4)$ for $0t60$ using the MATLAB

routine ode$45.$ Call the new function LAB$04$ex$1.$

Let $t,Y($note the upper case $Y)$ be the output of ode $45$ and $y$ and $v$ the unknown functions.

Use the following commands to define the ODE:

function $dYdt=f(t,Y)$

$y=Y(1)$;$v=Y(2)$;

$dYdt=[v$;$-3**sin(t)-3**v-6**y]$;

Plot $y(t)$ and $v(t)$ in the same window $($do not use subplot$),$ and the phase plot showing $v$ vs

$y$ in a separate window.

Add a legend to the first plot. $($Note: to display $v(t)=y^{\text{'}}(t),$ use $\text{'}v(t)=y^{\text{'}}\text{'}(t)\text{'}).$

Add a grid. Use the command ylim $([-2\mathrm{.}8,2\mathrm{.}8])$ to adjust the $y-$limits for both plots. Adjust

the $x-$limits in the phase plot so as to reproduce the pictures in Figure $7.$

Figure $7$: Time series $y=y(t)$ and $v=v(t)=y^{\text{'}}(t)($left$),$ and phase plot $v=y^{\text{'}}$ vs$.y$ for $(4).to.=0$;$tf=20$;$yo.=[10$;$60]$;

$a=\mathrm{.}8$;$b=\mathrm{.}01$;$c=\mathrm{.}6$;$d=\mathrm{.}1$;

$[t,y]=$ode$45($@$f,[to.,tf],yo.,[],a,b,c,d)$;

$u1=y($:$,1)$;$u2=y($:$,2)$;$,\%$ y in output has $2$ columns corresponding to u$1$ and u$2$

figure$(1)$;

subplot$(2,1,1)$; plot$($t$,$u$1,\text{'}$b$-+\text{'})$; ylabel$(\text{'}$u$1\text{'})$;

subplot$(2,1,2)$; plot$($t$,$u$2,\text{'}$ro$-\text{'})$; ylabel$(\text{'}$u$2\text{'})$;

figure$(2)$

plot$($u$1,$u$2)$; axis square; xlabel$(\text{'}$u$\_1\text{'})$; ylabel$(\text{'}$u$\_2\text{'})$; $\%$ plot the phase plot

$\%---------------------------------------------------------$

function dydt $=f(t,y,a,b,c,d)$

$u1=y(1)$;$u2=y(2)$;

dydt $=[a^{**}u1-b^{**}u{1}^{**}u2$;$-c^{**}u2+d^{**}u{1}^{**}u2]$;

end