Consider an autonomous system of $n$ coupled first order differential equations in a vector form:

$\frac{du}{dt}=f(u),u(t_0)=u_0,$

where $u=(u_1,u_2,...,u_n)$ and $f=(f_1,f_2,...,f_n).$

An efficient way to solve this system $($compute $u_{n+1}$ at $t=t_n+t),$ using the RK$4$ scheme is to

recast the scheme in autonomous vector form:

$k_1=tf(u_n)$

$k_2=tf(u_n+\frac{1}{2}{k}_1)$

$k_3=tf(u_n+\frac{1}{2}{k}_2)$

$k_4=tf(u_n+k_3)$

$u_{n+1}=u_n+\frac{1}{6}(k_1+2k_2+2k_3+k_4),$ for $n=0,1,2,...$

u$=2($x$)^(1)($u $$ ln x$)+($x$)^(1),$ u$(1)=0(1)$

Write a MatLab.m file to implement this algorithm to solve an autonomous first order system of equations. The vectors f and u$0,$ the step size $$t and the number of steps should be inputs to the function $1.$

Given that the exact solution of $(1)$ is yexact $=($ln x$)2+$ ln x$.(2).$

Use your MatLab function numerically determine how the error at x $=1\mathrm{.}8$ depends on h$.$ Choosing suitable values for step$-$size h$,$ plot log $|$error$|$ versus log h$.$ Use your plot to determine a relationship between the error and h$.$ How accurate is the method? Is this what you would

expect? Please discuss.