$1$ Use the recursion $x(t+t)=Bx(t)$ in writing a program that computes

$x(t),x(2t),$dots,$x(nt)$ given $x(0).$

$2($a$)$ For a matrix A and real number $h,$ write a program that finds

$S_n=I+Ah+\frac{A^2{h}^2}{2}+$dots$+\frac{A^n{h}^n}{n!}.$

$($b$)$ For the differential equation $x^{}=Ax,$ show that $x(t+h)=e^{Ah}x(t).$

$($Hint: Use the fact that $e^{A(s+t)}=e^{As}{e}^{At}.)$

$($c$)$ Use $($a$)$ and the recursion in $($b$)$ to estimate $x(h),x(2h),$dots,$x(nh)$

given $x(0).$

Part A

$3$ For the compartment diagram shown below,

$($a$)$ Find the transfer matrix A$.$

$($b$)$ If $x(0)=[100,250,80],$ find $x(1)$ if material is transferred once a

day.

$($c$)$ Find $x(1)$ if material is transferred hourly.