$\%\%$ Project Five

$\%[$YOUR NAME$]$

$\%[$Any Collaborators$]$

$\%\%$ Instructions

$\%$ RUN THE CODE TO CONFIRM THAT IT WORKS

$\%$ INCLUDE YOUR ANSWER TO QUESTION $1$ AS A COMMENT

$\%$ ADD AND MODIFY THE CODE WHERE INDICATED

$\%$ INCLUDE YOUR ANSWER TO QUESTION $2$ AS A COMMENT

$\%$ ADD AND MODIFY THE CODE WHERE INDICATED

$\%$ INCLUDE YOUR ANSWER TO QUESTION $3,$ ALONG WITH ANY SUPPORTING MATERIALS,

$\%$ AS A COMMENT

$\%$ USE THE Publish TAB TO SAVE YOUR RESULTS AS A PDF

$\%\%$ Initialize

$\%$ These commands clear the workspace and close any figures, to make sure

$\%$ that previous versions of the code do not effect the current run.

$\%$ They are not always necessary, and may noticably slow large progams.

clear all;

close all;

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

and 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.$)$

$\%$

$\%[$INSERT ANSWER HERE$]$

$\%\%$ Solve the ODE for omega $=2$

$\%[$INSERT MODIFIED CODE TO GRAPH THE SOLUTION OF THE ODE WITH omega $=2$ IN

$\%$ Figure$(2)$ HERE$]$

$\%\%$ Question $2$

$\%$ How has the solution changed? $($Be as exact as possible. You may wish to

$\%$ use the ZOOM feature in the Figures.$)$

$\%$

$\%[$INSERT ANSWER HERE$]$

$\%\%$ Solve the ODE for omega $=2\mathrm{.}6$

$\%[$INSERT MODIFIED CODE TO GRAPH THE SOLUTION OF THE ODE WITH omega $=2\mathrm{.}6$ IN

$\%$ Figure$(3)$ HERE$]$

$\%\%$ Question $3$

$\%$ How has the solution changed compared to the previous two solutions?

$\%$ What do you expect will happen as omega continues to increase? $($Be as

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

$\%$ If you run any simulations to confirm your hypothesis, be sure to include

$\%$ them in the report.$)$

$\%$

$\%[$INSERT ANSWER HERE$]$

$\%\%$ Terminate

$\%$ SOME students running SOME versions on SOME platforms have had issues

$\%$ with the last table or figure not appearing in the PUBLISH document.

$\%$ This workaround appears to fix it for most, but be aware and always check

$\%$ your results before submitting them for grade.

term $=$ true;

$\%\%$ Functions

function dYdt $=$ f$($t$,$Y$,$param$)$

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

omega$0=$ param$(1)$; c $=$ param$(2)$; omega $=$ param$(3)$;

dYdt $=[$ v ; cos$($omega$*$t$)-$omega$0^2*$y$-$c$*$v $]$;

end

$($answer the questions specific as possible using the results from the graphs produced and also fix the code$)$