COMPLETE THIS QUESTION USING THIS IN COMMAND WINDOW:

clear all;

f $=$ @$($x$,$y$)2*$x$+$y;

figure;

dirfield$($f$,-3$:$\mathrm{.}3$:$3,-4$:$\mathrm{.}4$:$4)$;hold on;

x $=-3$:$0\mathrm{.}1$:$3$;

y$1=2*$exp$($x$)-($x$+1)$;

y$2=2*$x$+1$;

y$3=$ cos$($x$)$;

plot$($x$,$y$1,\text{'}$r$\text{'})$;hold on;

plot$($x$,$y$2,\text{'}$g$\text{'})$;hold on;

plot$($x$,$y$3,\text{'}$b$\text{'})$;hold off;

PLEASE USE THIS GIVEN TO WRITE SCRIPT DIRFIELD:

function dirfield$($f$,$tval,yval$)$

$\%$ dirfield$($f$,$ t$1$:dt:t$2,$ y$1$:dy:y$2)$

$\%$

$\%$ plot direction field for first order ODE y$\text{'}=$ f$($t$,$y$)$

$\%$ using t$-$values from t$1$ to t$2$ with spacing of dt

$\%$ using y$-$values from y$1$ to t$2$ with spacing of dy

$\%$

$\%$ f is an @ function, or an inline function,

$\%$ or the name of an m$-$file with quotes.

$\%$

$\%$ Example: y$\text{'}=-$y$^2+$ t

$\%$ Show direction field for t in $[-1,3],$ y in $[-2,2],$ use

$\%$ spacing of $\mathrm{.}2$ for both t and y:

$\%$

$\%$ f $=$ @$($t$,$y$)-$y$^2+$t;

$\%$ dirfield$($f$,-1$:$\mathrm{.}2$:$3,-2$:$\mathrm{.}2$:$2)$;

$[$tm$,$ym$]=$meshgrid$($tval$,$yval$)$;

dt $=$ tval$(2)-$ tval$(1)$;

dy $=$ yval$(2)-$ yval$(1)$;

fv $=$ vectorize$($f$)$;

if isa$($f$,$'function$\_$handle'$)$

fv $=$ eval$($fv$)$;

end

yp$=$feval$($fv$,$tm$,$ym$)$;

s $=1./$max$(1/$dt$,$abs$($yp$)./$dy$)*0\mathrm{.}35$;

h $=$ ishold;

quiver$($tval$,$yval,s$,$s$.*$yp$,0,\text{'}.$r$\text{'})$; hold on;

quiver$($tval$,$yval,$-$s$,-$s$.*$yp$,0,\text{'}.$r$\text{'})$;

if h

hold on

else

hold off

end

axis$([$tval$(1)-$dt$/2,$tval$($end$)+$dt$/2,$yval$(1)-$dy$/2,$yval$($end$)+$dy$/2])$

Question Translation:

Consider the differential equation

$\frac{dy}{dx}=2x+y.$

$($a$)$ Write the expression of the explicit solution $y$ that passes through $(0,-2).$

You must show detailed computations in your report.

$($b$)$ Write the expression of the explicit solution $y$ that passes through $(-1,3).$

$($c$)$ Write the expression of the explicit solution $y$ that passes through $(-2,0).$

$($d$)$ Define the three solutions you find as $y1,y2,$ and $y3$ in Project$2\_$Q$1.$m$.$

Then use the script to obtain the direction field and plot the solutions in the

direction field. What can you say about the solutions $x+?$ How about

$x-?$