I have a working matlab code $-\%$ Parameters

M $=1$;

K $=3$;

B $=0\mathrm{.}5$;

x$0=2$;

v$0=0$;

t$0=0$;

tend $=15$;

h $=0\mathrm{.}01$;

$\%$ Function for Euler equation solution

f$1=$ @$($t$,$ x$1,$ x$2)$ x$2$;

f$2=$ @$($t$,$ x$1,$ x$2)-($B$/$M$)*$ x$2-($K$/$M$)*$ x$1$;

$\%$ Euler method

tn$\_$euler $=$ t$0$:h:tend;

x$1\_$euler $=$ zeros$($size$($tn$\_$euler$))$;

x$2\_$euler $=$ zeros$($size$($tn$\_$euler$))$;

x$1\_$euler$(1)=$ x$0$;

x$2\_$euler$(1)=$ v$0$;

fprintf$(\text{'}$All Euler steps

$\text{'})$;

for i $=1$:length$($tn$\_$euler$)-1$

t $=$ tn$\_$euler$($i$)$;

x$1=$ x$1\_$euler$($i$)$;

x$2=$ x$2\_$euler$($i$)$;

x$1\_$euler$($i$+1)=$ x$1+$ h $*$ f$1($t$,$ x$1,$ x$2)$;

x$2\_$euler$($i$+1)=$ x$2+$ h $*$ f$2($t$,$ x$1,$ x$2)$;

fprintf$(\text{'}\%$f $\%$f $\%$f

$\text{'},$ tn$\_$euler$($i$),$ x$1\_$euler$($i$),$ x$2\_$euler$($i$))$;

end

$\%$ Plotting numerical solution using Euler method

figure;

hold on;

plot$($tn$\_$euler, x$1\_$euler, 'linewidth', $2)$;

plot$($tn$\_$euler, x$2\_$euler, 'linewidth', $2)$;

plot$($tn$\_$euler, $-0\mathrm{.}5*$ x$2\_$euler $-3*$ x$1\_$euler, 'linewidth', $2)$;

legend$(\text{'}$Displacement$\text{'},$ 'Velocity', 'Acceleration'$)$;

title$(\text{'}$Solution using Euler method'$)$;

box on;

$\%$ Runge$-$Kutta $4$ method

x$1\_$rk$4=$ zeros$($size$($tn$\_$euler$))$;

x$2\_$rk$4=$ zeros$($size$($tn$\_$euler$))$;

x$1\_$rk$4(1)=$ x$0$;

x$2\_$rk$4(1)=$ v$0$;

fprintf$(\text{'}$All Runge Kutta $4$ steps

$\text{'})$;

for i $=1$:length$($tn$\_$euler$)-1$

t $=$ tn$\_$euler$($i$)$;

x$1=$ x$1\_$rk$4($i$)$;

x$2=$ x$2\_$rk$4($i$)$;

k$1=$ h $*$ f$1($t$,$ x$1,$ x$2)$;

l$1=$ h $*$ f$2($t$,$ x$1,$ x$2)$;

k$2=$ h $*$ f$1($t $+$ h$/2,$ x$1+$ k$1/2,$ x$2+$ l$1/2)$;

l$2=$ h $*$ f$2($t $+$ h$/2,$ x$1+$ k$1/2,$ x$2+$ l$1/2)$;

k$3=$ h $*$ f$1($t $+$ h$/2,$ x$1+$ k$2/2,$ x$2+$ l$2/2)$;

l$3=$ h $*$ f$2($t $+$ h$/2,$ x$1+$ k$2/2,$ x$2+$ l$2/2)$;

k$4=$ h $*$ f$1($t $+$ h$,$ x$1+$ k$3,$ x$2+$ l$3)$;

l$4=$ h $*$ f$2($t $+$ h$,$ x$1+$ k$3,$ x$2+$ l$3)$;

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

x$2\_$rk$4($i$+1)=$ x$2+($l$1+2*$l$2+2*$l$3+$ l$4)/6$;

fprintf$(\text{'}\%$f $\%$f $\%$f

$\text{'},$ tn$\_$euler$($i$),$ x$1\_$rk$4($i$),$ x$2\_$rk$4($i$))$;

end

$\%$ Plotting numerical solution using Runge$-$Kutta $4$ method

figure;

hold on;

ylim$([-6,6])$;

plot$($tn$\_$euler, x$1\_$rk$4,$ 'linewidth', $2)$;

plot$($tn$\_$euler, x$2\_$rk$4,$ 'linewidth', $2)$;

plot$($tn$\_$euler, $-0\mathrm{.}5*$ x$2\_$rk$4-3*$ x$1\_$rk$4,$ 'linewidth', $2)$;

legend$(\text{'}$Displacement$\text{'},$ 'Velocity', 'Acceleration'$)$;

title$(\text{'}$Solution using Runge$-$Kutta $4$ method'$)$;

box on; And I need to implement this to it$,$ Turn off the sliding friction and set h $=0\mathrm{.}1,$ but keep all other parameters the same.

Integrate using the Euler Method. You will see that the oscillator is now unstable.

As one shrinks h$,$ the oscillations become more convergent to steady state, however,

ideal behavior $($y$($t$)=$ y$($t $+$ T$),$ where T is the period of the oscillation$)$ requires

infinite numerical precision.

Show what the behavior looks like with the IEM and the RK$4$ method for h $=0\mathrm{.}1$

$[$DLV x $2].$

Find the limit of h for the Euler Method $($EM$)$ such that the oscillations increase by

less than $1\%$ over a period of $1$ oscillation, i$.$e$.,$ y$($t $+$ T$)<=1\mathrm{.}01*$ y$($t$)[$DLV$].$

For part II$,$ you may have to change the Integration$\_$Time to examine the stability.