Reaction kinetics for a bioreactor is given by:

$\frac{ds}{dt}=r_{1}s-\frac{r_1}{r_2}{s}^2$

$\frac{dx}{dt}=r_{3}s$

$\frac{dy}{dt}=r_{4}x$

$s=$ substrate concentration $(\frac{M}{L^3})$

$x=$ biomass concentration $(\frac{M}{L^3})$

$y=$ byproduct concentration

Rate constants are determined by experiments and found to be $r_1=12\mathrm{.}9,r_2=0\mathrm{.}89,r_3=1\mathrm{.}67,$ and $r_4=0\mathrm{.}75.$ Initial conditions are $s(0)=0\mathrm{.}05,x(0)=0,$ and $y(0)=0.$

$($a$)$ List the methods that you can use to solve this problem.

$($b$)$ Solve this problem with forward$-$Euler method in spreadsheet between $0t2$ with $t=0\mathrm{.}01.$ Plot $s,x,$ and y versus time. $($Not VBA programming$)$

$($c$)$ Using VBA program from practice final exam, solve this problem using $4^{th}-$order RK method. Plot $s,x,$ and y vs time.