We can now apply Heun's method to our system in Problem #$1,$ where the dependent variable is now a vector of unknowns Y $=($x$1,$ x$2,$ v$1,$ v$2).$

Our system of four ODEs is

x$1$x$2$v$1$v$2$

$=$ Y$=$ F$($x$1,$ x$2,$ v$1,$ v$2)$

where the vector F represents the right hand sides of the ODEs from problem #$1.$ For ease, we will write this as

Y$($t$)=$ F$($Y$($t$))$

At each time tn$,$ the vector Yn contains the displacements and velocities. Heun's method would first estimate

Y$*$n$+1=$ Yn $+$ h F $($Yn$)$

It would then produce a second, more accurate estimate using

Yn$+1=$ Yn $+$ h$$

$12$

$[$F$($Yn$)+$ F$($Y$*$n$+1)].$

Run Heun's method for the system given in Problem #$1.$

What is the maximum positive displacement achieved by mass $2$ on the time interval $[0,20]?$