The general solution to the differential equation dy

dx $=3$x $1$x$2+8$ cos$($x$)$ is

y$($x$)=3$

$2$ x$21$

$3$ x$3+8$ sin$($x$)+$ C with y$(0)=$ C$.$

The goal of this exercise is to write a script file to plot the solutions to the differential equation

in the interval $0<=$ x $<=9,$ with initial conditions y$(0)=20,40,60.$

The script file should have the structure script$+$function $($similarly to the M$-$file myplot$2.$m

Example $2).$ Call the file ex$5.$m$.$ The function that defines y$($x$)$ must be included in the same file

$($note that the function defining y$($x$)$ will have two input arguments: x and C$).$

Your M$-$file should have the following structure $($fill in all the $??$ with the appropriate commands$)$:

x $=??$ ; $\%$ define the vector x in the interval $[0,9]$

y$1=$ f $(??)$ ; $\%$ compute the solution with C $=20$

y$2=$ f $(??)$ ; $\%$ compute the solution with C $=40$

y$3=$ f $(??)$ ; $\%$ compute the solution with C $=60$

plot $(??)\%$ plot the three solutions with different line $-$ styles

title $(??)\%$ add a title

legend $(??)\%$ add a legend

function y $=$ f $($x $,$ C $)$

y $=??\%$ fill $-$ in with the expression for the general solution

end

Plot the graphs in the same window and use different color and$/$or line$-$styles for each graph. To

plot the graphs in the same window you can use the command hold on or use the plot command

similarly to Exercise $4.$

Add the title $$Solutions to dy$/$dx $=3$x $1$x$2+8$ cos$($x$).$

Add a legend with the list of C values used for each graph.

$($Type help plot for a list of the different line$-$styles, and help legend for help on how to add a

legend.$)$ Include both the M$-$file and the plot in your report.

NOTE: the only output of the script file should be the graph of the three curves. Make sure you

use enough points so that the curves are nice and smooth