Hey please help me solve and put this in Matlab Thanks here is the code that I am supposed to use:

$\%03/04/2024$

$\%$ Solve a $2$nd order ODE example

$\%$ y$\text{'}\text{'}-2$y$\text{'}+$ y $=$ e$^$t$/(1+$t$^2)$

tspan $=0$:$0\mathrm{.}01$:$10$;

y$0=[1$;$0]$;

$[$t$,$y$]=$ ode$45($@ode$1,$ tspan, y$0)$;

figure$(1)$

plot$($t$,$y$($:$,1))$

$\%$ exact solution

ye $=$ @$($t$)$ exp$($t$)-$exp$($t$).*$t$-$exp$($t$).*(-2*$atan$($t$).*$t $+$ log$($t$.^2+1))/2$;

hold on

plot$($t$,$ye$($t$),\text{'}--\text{'},$'linewidth',$2)$

$\%$

$\%$ max$($abs$($ye$($t$)-$y$($:$,1)))$

$\%\%$ In class assignment:

$\%$ Solve following differential equations using MAPLE and MATLAB and compare

$\%$ your results.

$\%\%($a$)$ y$\text{'}\text{'}+2*$y$\text{'}+5*$y$=4*$exp$(-$t$)*$cos$(2*$t$),$ y$(0)=1,$ y$\text{'}(0)=0$

tspan $=0$:$0\mathrm{.}01$:$10$;

y$0=[1$;$0]$;

$[$t$,$y$]=$ ode$45($@ode$2,$ tspan, y$0)$;

figure$(2)$

plot$($t$,$y$($:$,1))$

$\%$ exact solution

ye $=$ @$($t$)$ exp$(-$t$).*(2*$t$.*$sin$(2*$t$)+$ sin$(2*$t$)+2*$cos$(2*$t$))/2$;

hold on

plot$($t$,$ye$($t$),\text{'}--\text{'},$'linewidth',$2)$

$\%\%($b$)$ y$\text{'}\text{'}\text{'}+4*$y$\text{'}=$ t$,$ y$(0)=$y$\text{'}(0)=0,$ y$\text{'}\text{'}(0)=1$

tspan $=0$:$0\mathrm{.}01$:$10$;

y$0=[0$;$0$;$1]$;

$[$t$,$y$]=$ ode$45($@ode$3,$ tspan, y$0)$;

figure$(3)$

plot$($t$,$y$($:$,1))$

$\%$ exact solution

ye $=$ @$($t$)$ t$.^2/8+3/16-(3*$cos$(2*$t$))/16$;

hold on

plot$($t$,$ye$($t$),\text{'}--\text{'},$'linewidth',$2)$

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

function dydt $=$ ode$1($t$,$y$)$

dydt $=[$y$(2)$; $2*$y$(2)-$y$(1)+$exp$($t$)./(1+$t$.^2)]$;

end

function dydt $=$ ode$2($t$,$y$)$

dydt $=[$y$(2)$; $-2*$y$(2)-5*$y$(1)+4*$exp$(-$t$)*$cos$(2*$t$)]$;

end

function dydt $=$ ode$3($t$,$y$)$

dydt $=[$y$(2)$; y$(3)$; $-4*$y$(2)+$t$]$;

endUse the attached journal article $-$ "Chaos in a Three$-$Species Food Chain" by Hastings

and Powell $($Ecology $,$ Jun., $1991)$ to reproduce FIG. $2$ and FIG. $3.$

d$($x$)/($d$)$t$=$x$(1-$x$)-$f$\_(1)($x$)$y

d$($y$)/($d$)$t$=$f$\_(1)($x$)$y$-$f$\_(2)($y$)$z$-$d$\_(1)$y

d$($z$)/($d$)$t$=$f$\_(2)($y$)$z$-$d$\_(2)$z

f$\_($i$)($u$)=$a$\_($i$)($u$)/(1+$b$\_($i$)$u$)$Use the attached journal article $-$ "Chaos in a Three$-$Species Food Chain" by Hastings

and Powell $($Ecology $,$ Jun., $1991)$ to reproduce FIG. $2$ and FIG. $3.$

$d\frac{x}{d}t=x(1-x)-f_1(x)y$

$d\frac{y}{d}t=f_1(x)y-f_2(y)z-d_{1}y$

$d\frac{z}{d}t=f_2(y)z-d_{2}z$

$f_i(u)=a_i\frac{u}{1+b_{i}u}$