Consider the system of first order nonlinear ODEs called the Lorenz equations:

x$`\_1=$ o$($x$\_2-$ x$\_1)$

x$`\_2=$ x$\_1($p $-$ x$\_3)-$ x$\_2$

x$`\_3=($x$\_1)($x$\_2)-$ B$($x$\_3)$

where

x$`\_1=[$dx$\_1/$dt$],$ x$`\_2=[$dx$\_2/$dt$],$ x$`\_3=[$dx$\_3/$dt$]$

and

o $=10$

p $=28$

B $=8/3$

Solve the system of ODEs using Matlab's built$-$in function ode$45$

Interval: $0<=$ t $<=50$

Initial condition: x$\_1($t $=0)=1,$ x$\_2($t $=0)=1,$ x$\_3($t $=0)=2.$

set known variables

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

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

assign value to sigma

assign value to rho

assign value to beta

define ode function to update solution

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

name: dxdt

inputs: t $($scalar$)$ and x $(1$d array with $3$ entries$)$ and

ouptut: the rate of change of x $(1$d array with $3$ entries$)$

check dxdt function

to do this carry out the following steps

$-$ assign the value $1$ to a variable called t$\_$check

$-$ assign the values $1,2$ and $3$ to a row vector called x$\_$check

$-$ call the function dxdt$,$ pass in t$\_$check and x$\_$check and store the returned result in dxdt$\_$check

assign t$\_$check

assign x$\_$check

call function dxdt and pass in t$\_$check and x$\_$check, store the returned

value in dxdt$\_$check

set the initial condition

store the initial values for x$\_1($t $=0),$ x$\_2($t $=0)$ and x$\_3($t$=0)$ in a row vector called x$0$

solve ode using ode$45$ store the returned solution in t and x

plot results

plot the values of the third column of x over the first column of x as a solid red line with the line width $4$

customize the plot as follows:

$-$ switch on grid

$-$ swith font size to $20$

$-$ set the x$-$axis label to $\text{'}$x$\_1\text{'}$

$-$ set the y$-$axis label to $\text{'}$x$\_3\text{'}$

MATLAB code please