Please fix my matlab code so that Improved Euler Method is correct $\%$ Define the parameters

y$0=1$;

h$\_$values $=[1/8,1/16,1/32,1/64]$;

t$\_$end $=20$;

$\%$ Initialize time vector

t $=0$:max$($h$\_$values$)$:t$\_$end;

$\%$ Preallocate arrays

n $=$ length$($t$)$;

y$\_$improved$\_$euler $=$ zeros$($length$($h$\_$values$),$ n$)$;

y$\_$rk$4=$ zeros$($length$($h$\_$values$),$ n$)$;

exact$\_$solution $=$ zeros$(1,$ n$)$;

$\%$ Initial conditions

y$\_$improved$\_$euler$($:$,1)=$ y$0$;

y$\_$rk$4($:$,1)=$ y$0$;

exact$\_$solution$(1)=$ y$0$;

$\%$ Define the differential equation function

dydt $=$ @$($t$,$ y$)$ y $-$ y$.^2+1\mathrm{.}14*$ cos$($exp$($t$/2))$;

$\%$ Improved Euler's method for each time step

for j $=1$:length$($h$\_$values$)$

h $=$ h$\_$values$($j$)$;

for i $=1$:n$-1$

k$1=$ h $*$ dydt$($t$($i$),$ y$\_$improved$\_$euler$($j$,$ i$))$;

k$2=$ h $*$ dydt$($t$($i$)+$ h$,$ y$\_$improved$\_$euler$($j$,$ i$)+$ k$1)$;

y$\_$improved$\_$euler$($j$,$ i$+1)=$ y$\_$improved$\_$euler$($j$,$ i$)+0\mathrm{.}5*($k$1+$ k$2)$;

end

end

$\%$ RK$4$ method for each time step

for j $=1$:length$($h$\_$values$)$

h $=$ h$\_$values$($j$)$;

for i $=1$:n$-1$

k$1=$ h $*$ dydt$($t$($i$),$ y$\_$rk$4($j$,$ i$))$;

k$2=$ h $*$ dydt$($t$($i$)+$ h$/2,$ y$\_$rk$4($j$,$ i$)+$ k$1/2)$;

k$3=$ h $*$ dydt$($t$($i$)+$ h$/2,$ y$\_$rk$4($j$,$ i$)+$ k$2/2)$;

k$4=$ h $*$ dydt$($t$($i$)+$ h$,$ y$\_$rk$4($j$,$ i$)+$ k$3)$;

y$\_$rk$4($j$,$ i$+1)=$ y$\_$rk$4($j$,$ i$)+(1/6)*($k$1+2*$k$2+2*$k$3+$ k$4)$;

end

end

$\%$ Exact solution using numerical integration

opts $=$ odeset$(\text{'}$RelTol$\text{'},1$e$-8,$ 'AbsTol', $1$e$-8)$;

$[$t$\_$exact, y$\_$exact$]=$ ode$45($dydt$,$ t$,$ y$0,$ opts$)$;

$\%$ Plot Improved Euler's method approximations

figure;

hold on;

for j $=1$:length$($h$\_$values$)$

plot$($t$,$ y$\_$improved$\_$euler$($j$,$:$),\text{'}-\text{'},$ 'LineWidth', $1\mathrm{.}5)$;

end

plot$($t$\_$exact, y$\_$exact, $\text{'}$k$--\text{'},$ 'LineWidth', $1\mathrm{.}5)$;

title$("$Improved Euler's Method Approximation"$)$;

xlabel$(\text{'}$t$\text{'})$;

ylabel$(\text{'}$y$\text{'})$;

legend$(\text{'}$h $=1/8\text{'},\text{'}$h $=1/16\text{'},\text{'}$h $=1/32\text{'},\text{'}$h $=1/64\text{'},$ 'Exact Solution'$)$;

grid on;

$\%$ Plot RK$4$ method approximations

figure;

hold on;

for j $=1$:length$($h$\_$values$)$

plot$($t$,$ y$\_$rk$4($j$,$:$),\text{'}-\text{'},$ 'LineWidth', $1\mathrm{.}5)$;

end

plot$($t$\_$exact, y$\_$exact, $\text{'}$k$--\text{'},$ 'LineWidth', $1\mathrm{.}5)$;

title$("$RK$4$ Method Approximation"$)$;

xlabel$(\text{'}$t$\text{'})$;

ylabel$(\text{'}$y$\text{'})$;

legend$(\text{'}$h $=1/8\text{'},\text{'}$h $=1/16\text{'},\text{'}$h $=1/32\text{'},\text{'}$h $=1/64\text{'},$ 'Exact Solution'$)$;

grid on;

$\%$ Discussion of results

disp$("$Results Discussion:"$)$;

disp$("$Improved Euler's method generally improves accuracy with smaller h values due to its higher$-$order approximation."$)$;

disp$("$RK$4$ method consistently shows higher accuracy compared to Improved Euler's method across all tested h values."$)$;

disp$("$Exact solution closely matches RK$4$ method, validating its accuracy."$)$;