Consider the rst order initial value problem:

dy

dx $=$y$+$y$^2($sin$($x$)$cos$($x$)),$

y $(0)=0\mathrm{.}1$

$($a$)$ Determine the exact solution of the given initial value problem $($manually$).$ Please show

all your working.

$($b$)$ Estimate the value of y $(1)$ using Euler's Method with step size h $=0\mathrm{.}2($manually$),$ correct

to $4$ decimal places. Show all your working.

$($c$)$ Write a MATLAB function Euler$1.$m to solve the initial value problem

dy

dx $=$y$+$y$2($sin$($x$)$cos$($x$)),$

y $(0)=$ y$0$

numerically with Euler's method over the general interval x in $[0,$b$]$ for the general initial

condition y$(0)=$ y$0,$ using n regularly spaced points for the x grid. The input variables for

Euler$1$ should be b$,$ y$0$ and n$.$ Your code should generate a plot of the solution y$($x$),$while

automatically inserting the label y for the y$-$axis, the label x for the x axis, and the

title Numerical Solution of Question $1$ for your plot.

Next, use your MATLAB function Euler$1.$m to generate and plot the numerical solution

of this initial value problem for the speci ed initial condition y$(0)=0\mathrm{.}1$ over the interval

x in $[0,2],$ using n $=101$ points in your x grid. Make sure to include the plot in your

submission.

$($d$)$ Using appropriate MATLAB commands, overlay a plot of the exact solution found in Part

$($a$)$ onto the plot obtained in Part $($c$).$ The curve for the exact solution should be a dashed

line in your plot.

$($e$)$ State whether the numerical solution found in Part $($c$)$ is an overestimate or underestimate

of the actual solution