$($a$)$ Determine the Improved Euler's approximation for $\backslash ($ N$=90,$ N$=900\backslash )$ and $\backslash ($ N$=9000\backslash ).$ Fill in the following table with the values of the approximations, errors and ratios of consecutive errors at $\backslash ($ t$=2\mathrm{.}25\backslash ).$ Two values have already been entered to help you check your results. Recall that the exact solution to the ODE is $\backslash ($ y$($t$)=-4$ e$^\{0\mathrm{.}5$ t$\}\backslash ).$ Include the table in your report, as well as the MATLAB commands used to find the entries.

$($b$)$ Examine the last column. How does the ratio of the errors relate to the number of steps used? Your answer to this question should confirm the fact that Improved Euler's method is a "second$-$order" method. That is$,$ every time the step size is decreased by a factor $\backslash ($ k $\backslash ),$ the error is reduced $($approximately$)$ by a factor of $\backslash ($ k$^\{2\}\backslash ).$

Note: Since Euler's method is only of the $1$st order, the Improved Euler's method is more efficient $($hence the "improved"$).$