EXERCISES

$($a$)$ Consider the modified problem

$($d$^(2)$y$)/($dt$^(2))+8$y$^(2)($dy$)/($dt$)+7$y$=8$sint$,$ with $,$y$(0)=-1,($dy$)/($dt$)(0)=0$

The ODE $(7)$ is very similar to $(4)$ except for the y$^(2)$ term in the left$-$hand side. Because of the

factor y$^(2)$ the ODE $(7)$ is nonlinear, while $(4)$ is linear. There is however very little to change

in the implementation of $(4)$ to solve $(7).$ In fact, the only thing that needs to be modified is

the ODE definition.

Modify the function defining the ODE in LABO$4$ex$1.$m$.$ Call the revised file LABO$4$ex$2.$m$.$ The

new function M$-$file should reproduce the pictures in Fig $8.$

Include in your report the changes you made to LABO$4$ex$1.$m to obtain LABO$4$ex$2.$m$.$

Figure $8$: Time series y$=$y$($t$)$ and v$=$v$($t$)=$y$^(\text{'})($t$)($left$),$ and phase plot v$=$y$^(\text{'})$ vs$.$ y for $(7).$

$($b$)$ Compare the output of Figs $7$ and $8.$ Describe the changes in the behavior of the solution in

the short term.

$($c$)$ Compare the long term behavior of both problems $(4)$ and $(7),$ in particular the amplitude of

oscillations.

$($d$)$ Modify LAB$04$ex$2.$m so that it solves $(7)$ using Euler's method with N$=600$ in the interval

tuse the file euler.m from LAB $3$ to implement Euler's method; do not delete the

lines that implement ode$45).$ Let $[$te$,$ Ye$]$ be the output of euler, and note that Ye is a ma$-$

trix with two columns from which the Euler's approximation to y$($t$)$ must be extracted. Plot

the approximation to the solution y$($t$)$v$($t$)$ nor the phase

plotN $?$

ex$\_$with$\_2$eqs.m

t$0=0$; tf $=20$; y$0=[10$;$60]$;

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

$[$t$,$y$]=$ ode$45($@f$,[$t$0,$tf$],$y$0,[],$a$,$b$,$c$,$d$)$;

u$1=$ y$($:$,1)$; u$2=$ 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$)$

u$1=$ y$(1)$; u$2=$ y$(2)$;

dydt $=[$ a$*$u$1-$b$*$u$1*$u$2$ ; $-$c$*$u$2+$d$*$u$1*$u$2]$;

end

ex$\_$with$\_$param

function ex$\_$with$\_$param

t$0=0$; tf $=3$; y$0=1$;

a $=1$;

$[$t$,$y$]=$ ode$45($@f$,[$t$0,$tf$],$y$0,[],$a$)$;

disp$([\text{'}$y$(\text{'}$ num$2$str$($t$($end$))\text{'})=\text{'}$ num$2$str$($y$($end$))])$

disp$([\text{'}$length of y $=\text{'}$ num$2$str$($length$($y$))])$

end

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

function dydt $=$ f$($t$,$y$,$a$)$

dydt $=-$a$*($y$-$exp$(-$t$))-$exp$(-$t$)$;

end