I desperatley need help with matlab problem.

I have attached my code and the current errors I$\text{'}$m encountering.Consider the following ODE:

$\frac{d^{2}y}{d{x}^2}+x^2\frac{dy}{dx}+y=4cos(x),$ for $,0x10,$ with $,y(0)=3,$ and $,y(10)=2$

Write a script that:

named ys$.$

finite difference approximations in your formulation.

b$)$ Solve for $y(x)$ using the finite difference method with $x=0\mathrm{.}5.$ Save result as a column vector named $ydx.$

c$)$ Solve for $y(x)$ using the finite difference method with $x=0\mathrm{.}25.$ Save result as a column vector named $ydx2.$

d$)$ Solve for $y(x)$ using the finite difference method with $x=0\mathrm{.}05.$ Save result as a column vector named $ydx3.$

e$)$ Compare the accuracy of your results for $($b$),($c$),$ and $($d$)$ by doing the following:

vectors of $lo{n}_a$ values. Note this means you will need to extract only the relevant $y$ values $($corresponding to the specified nodes$)$ from your solutions for $($b$),($c$)$ and $($d$)$ to use in this calculation.

Here is my code:

$\%$ Shooting Method

$\%$ Initial conditions

y$0=3$;

z$0\_$guess$1=0\mathrm{.}1$; $\%$ First guess for the slope at x$=0$

z$0\_$guess$2=0\mathrm{.}2$; $\%$ Second guess for the slope at x$=0$

$\%$ Define the system of ODEs within the script

odesys $=$ @$($x$,$ yz$)[$yz$(2)$; $4*$cos$($x$)-$ x$^2*$yz$(2)-$ yz$(1)]$;

$\%$ Specify the x values where the solution is to be evaluated

x$\_$eval $=$ linspace$(0,10,201)$; $\%201$ points from $0$ to $10$

$\%$ First shot

$[$x$1,$ yz$1]=$ ode$45($odesys$,$ x$\_$eval, $[$y$0$ z$0\_$guess$1])$;

$\%$ Second shot

$[$x$2,$ yz$2]=$ ode$45($odesys$,$ x$\_$eval, $[$y$0$ z$0\_$guess$2])$;

$\%$ Interpolate to find a better guess for z$(0)$

z$0\_$final $=$ interp$1([$yz$1($end$,1)$ yz$2($end$,1)],[$z$0\_$guess$1$ z$0\_$guess$2],2,$ 'linear', 'extrap'$)$;

$\%$ Final shot with interpolated initial slope

$[$x$\_$final, yz$\_$final$]=$ ode$45($odesys$,$ x$\_$eval, $[$y$0$ z$0\_$final$])$;

$\%$ Extract the final solution for y

yS $=$ yz$\_$final$($:$,1)$; $\%201$ elements

$\%$ Part $2$

function h $=$ solve$\_$finite$\_$difference$($dx$)$

$\%$ Constants

N $=1$; K $=1$; hbar $=-6$; h$0=3$; hn $=2$; xspan $=[0,10]$;

$\%$ Finite difference method

RHS $=-$N$*$dx$^2/$K$/$hbar;

$\%$ Internal nodes

n$\_$nodes $=$ xspan$($end$)/$dx $-1$;

b $=$ RHS$*$ones$($n$\_$nodes,$1)$;

b$(1)=$ b$(1)-$ h$0$;

b$($end$)=$ b$($end$)-$ hn;

A $=1*$diag$($ones$($n$\_$nodes$-1,1),-1)-2*$diag$($ones$($n$\_$nodes,$1),0)+1*$diag$($ones$($n$\_$nodes$-1,1),1)$;

h $=$ A$\backslash$b;

h $=[$h$0$; h; hn$]$;

end

$\%$ Solve the BVP using the finite difference method with different step sizes

y$\_$dx$1=$ solve$\_$finite$\_$difference$(0\mathrm{.}5)$;

y$\_$dx$2=$ solve$\_$finite$\_$difference$(0\mathrm{.}25)$;

y$\_$dx$3=$ solve$\_$finite$\_$difference$(0\mathrm{.}05)$;

function normEa $=$ compare$\_$accuracy$($ys$,$ y$\_$dx$1,$ y$\_$dx$2,$ y$\_$dx$3,$ delta$\_$x$1,$ delta$\_$x$2,$ delta$\_$xa$)$

$\%$ x$\_$output specifies the points where ys is calculated

x$\_$output $=0$:$0\mathrm{.}05$:$10$;

$\%$ Extract relevant y values from finite difference solutions

$\%$ Interpolate y$\_$dx$1$ to x$\_$output points

x$1=0$:delta$\_$x$1$:$10$;

y$1\_$interp $=$ interp$1($x$1,$ y$\_$dx$1,$ x$\_$output$)$;

$\%$ Interpolate y$\_$dx$2$ to x$\_$output points

x$2=0$:delta$\_$x$2$:$10$;

y$2\_$interp $=$ interp$1($x$2,$ y$\_$dx$2,$ x$\_$output$)$;

$\%$ Interpolate y$\_$dx$3$ to x$\_$output points

xa $=0$:delta$\_$xa:$10$;

ya$\_$interp $=$ interp$1($xa$,$ y$\_$dx$3,$ x$\_$output$)$;

$\%$ Calculate the relative errors

$\%$ Relative error for delta$\_$x$1$

err$1=$ abs$(($ys $-$ y$1\_$interp$)./$ ys$)$;

$\%$ Relative error for delta$\_$x$2$

err$2=$ abs$(($ys $-$ y$2\_$interp$)./$ ys$)$;

$\%$ Relative error for delta$\_$xa

erra $=$ abs$(($ys $-$ ya$\_$interp$)./$ ys$)$;

$\%$ Calculate the Euclidean norms of the error vectors

normEa $=[$norm$($err$1),$ norm$($err$2),$ norm$($erra$)]$;

end

$\%$ Calculate the Euclidean norms of the errors for different delta$\_$x values

normEa $=$ compare$\_$accuracy$($yS$,$ y$\_$dx$1,$ y$\_$dx$2,$ y$\_$dx$3,0\mathrm{.}5,0\mathrm{.}25,0\mathrm{.}05)$

$\%$ Display results

yS

y$\_$dx$1$

y$\_$dx$2$

y$\_$dx$3$

Errors:

Previous Assessment: $1$ of $5$ Tests Passed

Is the solution for the Shooting Method correct?

Is the solution for the Finite Difference Method delta $x=0\mathrm{.}5$ correct?

Variable y$\_$dx$1$ has an incorrect value.

Is the solution for the Finite Difference Method delta $x=0\mathrm{.}25$ correct?

Variable $yddx2$ has an incorrect value.

Is the solution for the Finite Difference Method delta $x=0\mathrm{.}05$ correct?

Variable y$\_$dx$3$ has an incorrect value.

Is the norm of the relative approxmiate error correct?

Variable normEa has an incorrect value.