Exercise $3\mathrm{.}1$

a$.$ Run Euler's Method again, this time for $h=0\mathrm{.}2.$ Plot the results in blue this time. $($Use $\text{'}b\text{'}$ instead of $\text{'}r\text{'}.)$ Look at the arrays $x$ and $y$ side$-$by$-$side by typing in

$>>x,y$

Copy the long table of data that you just produced into your Word document. What estimates did the Euler Method come up with for $y$ at $x=1$ and $x=2$ using this smaller value of $h?$ Now run Euler's Method for $h=0\mathrm{.}1$ and again for $h=0\mathrm{.}01,$ and plot the results in green $(\text{'}g\text{'})$ and cyan $(\text{'}c\text{'}),$ respectively. What are the estimates for $y$ at $x=1$ and $x=2$ for these values of $h?$ Write these down in your Word document, along with your graph with all four colored curves. $($You need not include the data tables for these last two values of $h.)$

b$.$ The differential equation given in $(2)$ is separable. Calculate the solution to the initial value problem by hand, and use MATLAB $($or a calculator$)$ to compute the actual values for $y$ at $x=1$ and $x=2.$ Make sure to include your manual work in your Word document. $($You can simply type it$.)$

c$.$ Does Euler's estimate appear to give better or worse estimates for the solution as $h$ decreases? Give a brief reasoning for why this might be the case. Does Euler's estimate appear to give better or worse estimates for the actual solution as you move farther fRom the initial

point?

Respond in MATLAB code please!