I need help with this matlab problem. I have attached my current code.

The deflection $y$ of a sailboat mast subjected to a wind force can be modeled with the following differential equation:

$\frac{d^{2}y}{d{z}^2}=\frac{f(z)}{2EI}(L-z{)}^2$

where $z$ is the position along the mast starting from $0$ at the base. The function $f(z)$ is the force of the wind transferred to the mast by the sail and can be modeled as

$f(z)=\frac{200z}{5+z}{e}^{\frac{-2z}{30}}$

parameter $L$ is the length of the mast.

Write a MATLAB function P$2\_$odesolve that will solve for deflection $y(z)$ and the slope $\frac{dy}{dz}$ given that the deflection and the slope are both equal to $0$ at the base of the mast with the following outputs $($in given order$)$:

Solutions using Euler's method as a $2-$column matrix $($store solution for deflection $y(z)$ in the first column and the solution for slope $\frac{dy}{dz}$ in the second column$)$

Solutions using Heun's method $($without corrector iteration$)$ as a $2-$column matrix $($store solution for deflection $y(z)$ in the first column and the solution for slope $\frac{dy}{dz}$ in the second column$)$

Solutions using Midpoint method as a $2-$column matrix $($store solution for deflection $y(z)$ in the first column and the solution for slope $\frac{dy}{dz}$ in the second column$)$

with the following input

vector defining the span of the mast with intial and final position values $(12$ row vector$)$

initial value vector for the deflection and the slope $(12$ row vector$)$

solver step size $h($scalar$)$ Function $($

$1$ function $[yE,yH,yM]=P2$ odesolve$($zspan$,y0,h)$

$2\%$type your code below, do not change the name of the function and the order of the variables$\%$

Code to call your function $(8)$

My code:

function $[$yE$,$ yH$,$ yM$]=$ P$2\_$odesolve$($zspan$,$ y$0,$ h$)$

$\%$ Constants

E $=1\mathrm{.}3$e$9$;

I $=0\mathrm{.}05$;

L $=$ zspan$(2)-$ zspan$(1)$;

$\%$ Number of steps

N $=$ round$(($zspan$(2)-$ zspan$(1))/$ h$)$;

$\%$ Initialize solution matrices

yE $=$ zeros$($N$+1,2)$;

yH $=$ zeros$($N$+1,2)$;

yM $=$ zeros$($N$+1,2)$;

$\%$ Initial conditions

yE$(1,$ :) $=$ y$0$;

yH$(1,$ :) $=$ y$0$;

yM$(1,$ :) $=$ y$0$;

$\%$ Function for f$($z$)$

f $=$ @$($z$)((200*$ z$)/(5+$ z$))*$ exp$(-2*$ z $/30)$;

$\%$ Euler's Method

for n $=1$:N

z $=$ zspan$(1)+($n$-1)*$ h;

y $=$ yE$($n$,1)$;

v $=$ yE$($n$,2)$;

yE$($n$+1,1)=$ y $+$ h $*$ v;

yE$($n$+1,2)=$ v $+$ h $*($f$($z$)/(2*$ E $*$ I $*($L $-$ z$)^2))$;

end

$\%$ Heun's Method

for n $=1$:N

z $=$ zspan$(1)+($n$-1)*$ h;

y $=$ yH$($n$,1)$;

v $=$ yH$($n$,2)$;

$\%$ Predictor step

y$\_$star $=$ y $+$ h $*$ v;

v$\_$star $=$ v $+$ h $*($f$($z$)/(2*$ E $*$ I $*($L $-$ z$)^2))$;

$\%$ Corrector step

yH$($n$+1,1)=$ y $+($h $/2)*($v $+$ v$\_$star$)$;

yH$($n$+1,2)=$ v $+($h $/2)*(($f$($z$)/(2*$ E $*$ I $*($L $-$ z$)^2))+($f$($z $+$ h$)/(2*$ E $*$ I $*($L $-($z $+$ h$))^2)))$;

end

$\%$ Midpoint Method

for n $=1$:N

z $=$ zspan$(1)+($n$-1)*$ h;

y $=$ yM$($n$,1)$;

v $=$ yM$($n$,2)$;

$\%$ Midpoint step

y$\_$mid $=$ y $+($h $/2)*$ v;

v$\_$mid $=$ v $+($h $/2)*($f$($z$)/(2*$ E $*$ I $*($L $-$ z$)^2))$;

$\%$ Full step

yM$($n$+1,1)=$ y $+$ h $*$ v$\_$mid;

yM$($n$+1,2)=$ v $+$ h $*($f$($z $+$ h $/2)/(2*$ E $*$ I $*($L $-($z $+$ h $/2))^2))$;

end

end