I need help with part of this matlab problem. The finite difference method is incorrect. I have attached the promt, my code, and the 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.

Code:

$\%$ 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 $2$: Finite Difference Method

$\%$ Define the ODE parameters

x$\_$end $=10$;

y$0=3$;

y$\_$end $=2$;

$\%$ Define the range of dx values

dx$\_$values $=[0\mathrm{.}5,0\mathrm{.}25,0\mathrm{.}05]$;

$\%$ Loop over the dx values

for i $=1$:length$($dx$\_$values$)$

dx $=$ dx$\_$values$($i$)$;

x $=0$:dx:x$\_$end;

N $=$ length$($x$)$;

$\%$ Initialize the matrix A for the linear system Ax $=$ b

A $=-2*$diag$($ones$($N$,1))+$ diag$($ones$($N$-1,1),1)+$ diag$($ones$($N$-1,1),-1)$;

A $=$ A $/$ dx$^2$;

$\%$ Initialize the vector b for the linear system Ax $=$ b

b $=4*$cos$($x$)\text{'}*$ dx$^2$;

$\%$ Apply boundary conditions

A$(1,$ :) $=0$; A$(1,1)=1$; b$(1)=$ y$0$;

A$($end$,$ :) $=0$; A$($end$,$ end$)=1$; b$($end$)=$ y$\_$end;

$\%$ Solve the linear system

y $=$ A$\backslash$b;

$\%$ Store the solution in the specified variable

if dx $==0\mathrm{.}5$

y$\_$dx$1=$ y;

elseif dx $==0\mathrm{.}25$

y$\_$dx$2=$ y;

elseif dx $==0\mathrm{.}05$

y$\_$dx$3=$ y;

end

end

$\%$ Define the nodes for comparison

nodes $=0$:$0\mathrm{.}5$:$10$;

$\%$ Extract the relevant y values

yS$\_$nodes $=$ interp$1($x$\_$final, yS$,$ nodes$)$;

$\%$ Initialize the error vectors

e$\_$dx$1=$ zeros$($length$($nodes$),1)$;

e$\_$dx$2=$ zeros$($length$($nodes$),1)$;

e$\_$dx$3=$ zeros$($length$($nodes$),1)$;

$\%$ Calculate the errors

for i $=1$:length$($nodes$)$

e$\_$dx$1($i$)=$ yS$\_$nodes$($i$)-$ interp$1(0$:$0\mathrm{.}5$:x$\_$end, y$\_$dx$1,$ nodes$($i$))$;

e$\_$dx$2($i$)=$ yS$\_$nodes$($i$)-$ interp$1(0$:$0\mathrm{.}25$:x$\_$end, y$\_$dx$2,$ nodes$($i$))$;

e$\_$dx$3($i$)=$ yS$\_$nodes$($i$)-$ interp$1(0$:$0\mathrm{.}05$:x$\_$end, y$\_$dx$3,$ nodes$($i$))$;

end

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

normE$\_$dx$1=$ norm$($e$\_$dx$1,2)$;

normE$\_$dx$2=$ norm$($e$\_$dx$2,2)$;

normE$\_$dx$3=$ norm$($e$\_$dx$3,2)$;

$\%$ Store the norms in a row vector

normEa $=[$normE$\_$dx$1,$ normE$\_$dx$2,$ normE$\_$dx$3]$;

$\%$ Display results

disp$(\text{'}$Norm of Errors:$\text{'})$;

disp$($normEa$)$;

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.