$\%\%$ Solve the ODE y$"+$c$*$y$\text{'}+$ omega$0^2*$y $=$ cos$($omega$*$t$)$ given y$(0)=0,$

y$\text{'}(0)=0$;

omega $=1\mathrm{.}4$;

omega$0=2$; c $=1$; omega $=1\mathrm{.}4$; $\%$ define parameters

param $=[$omega$0,$c$,$omega$]$; $\%$ convert parameters into form usable by ODE$45$

t$0=0$; y$0=0$; v$0=0$; Y$0=[$y$0$;v$0]$; tf $=50$; $\%$ define interval and

intial condition

options $=$ odeset$(\text{'}$AbsTol$\text{'},1$e$-10,$'RelTol',$1$e$-10)$; $\%$ adjust

tolerance$/$precision of ODE$45$

$[$t$,$Y$]=$ ode$45($@f$,[$t$0,$tf$],$Y$0,$options,param$)$;

y $=$ Y$($:$,1)$; v $=$ Y$($:$,2)$; $\%$ y in output has $2$ columns corresponding to y and

y$\text{'}$

figure$(1)$; $\%$ starts a new figure

plot$($t$,$y$,\text{'}$b$-\text{'})$; $\%$ graph solution

grid on; $\%$ display grid

$\%\%$ Question $1$

$\%$ What is the long term behavior of the solution? $($Be as exact as

$\%$ possible. You may wish to use the ZOOM feature in the Figures.$)$

$\%$