Consider the following ODE:

y$`=-2$xsin$($x$^2)$y

Solve the ODE using the fourth order Runge$-$Kutta $($RK$4)$ method:

k$1=$ F$($x$\_$i$,$y$\_$i$)$

k$2=$ F$($xi$+(1/2)($delta x$),$ y$\_$i$+(1/2)($k$1)($delta x$)$

k$3=$ F$($xi$+(1/2)($delta x$),$ y$\_$i$+(1/2)($k$2)($delta x$)$

k$4=$ F$($xi$+($delta x$),$ yi$+$k$3($delta x$)$

y$\_($i$+1)=$ y$\_$i $+(1/6)($delta x$)($k$1+2$k$2+2$k$3+$ k$4)$

where $$F$($x$\_$i$,$y$\_$i$)=$ y$`($x$\_$i$,$ y$\_$i$)=-$x$\_$i$($sin$[($x$\_$i$)^2]($y$\_$i$)$

Interval: $0<=$ x $<=5$

Number of points: n$\_$x $=100$

Initial condition: y$($x$=0)=$e

Hint: e$$is the mathematical constant e$$which is available in Maltab through the exp$()$ function. To get the value of e$$you can call the exp$()$function with input $1$: e $=$ exp$(1)$

discretize x

define start value of x: x$\_$s

define end value of x: x$\_$e

define number of points: n$\_$x

discretize x

compute the grid spacing dx

exact solution $($compute on finer discretization for visualization$)$

x$\_$vis $=$ linspace$($x$\_$s$,$x$\_$e$,1000)$;

y$\_$vis $=$ exp$($cos$($x$\_$vis.$^2))$;

preallocate array to store solution y with n$\_$x rows and $1$ column

define ode function to update solution

define an anonymous function $($function handle$)$ with the following properties

name: dydx

inputs: x and y $($both scalars$)$

ouptut: the rate of change of y $($scalar$)$

check dydx function

assign the value $2$ to x$\_$check

assign the value $1$ to y$\_$check

call function dydx and pass in x$\_$check and y$\_$check, store the returned

value in dydx$\_$check

solve the ode using RK$4$

apply initial condition

march forward and compute the solution using the RK$4$ method

plot results

plot y over x as a solid blue line with line width of $4$

plot y$\_$vis over x$\_$vis as a dashed red line with a line width of $4$ in the same plot

customize the plot as follows:

$-$ switch on grid

$-$ swith font size to $20$

$-$ set the x$-$axis label to $\text{'}$x$\text{'}$

$-$ set the y$-$axis label to $\text{'}$y$\text{'}$

$-$ add the legend entries $\text{'}$rk$4\text{'}$ and 'exact'

MATLAB Code Please