EXERCISES

$($a$)$ If you haven't already done so$,$ enter the following commands:

f$=$@$($t$,$y$)$ y;

t$=$linspace $(0,1\mathrm{.}25,100)$; $y=3**exp(t)$; $\%$define exact solution of the ODE $[$t$50,$y$50]=$euler$($f$,[0,1\mathrm{.}25],3,50)$; $\%$solve the ODE using Euler w$/50$ steps Determine the Euler's approximation for $N=500$ and $N=5000$ and fill in the following table with the values of the approximations $($ yN $($end$)),$ errors $(eN=y($end$)-$yN $($end$))$ and ratios $($eNprevious$/$eN$)$ of consecutive errors at $t=1\mathrm{.}25.$ Some of the values have already been entered based on the computations we did above.

Include the table in your report, as well as the MATLAB commands used to find the entries.

$\backslash$table$[[N,\backslash$table$[[$approximation$],[$yN$($end$)]],\backslash$table$[[$error$],[$eN$]],\backslash$table$[[$ratio$],[$eNprev$/$eN$]]],[5,9\mathrm{.}1553,1\mathrm{.}3158,\frac{N}{A}$