Problem $2(20$ points$).$ The logistic equation for batch microbial growth is given by the following equation:

$\frac{dx}{dt}=kx(1-\frac{x}{x_m})$

a$.$ Use the following approximations:

$x_m=Y_{\frac{x}{s}}{S}_0,x=Y_{\frac{x}{s}}(S_0-S)$

Prove that the logistic equation can be expressed as follows:

$\frac{dx}{dt}=kx\frac{S}{S_0}$

b$.$ By using the following relationship, develop an expression describing the variation of substrate concentration with time according to the logistic equation:

$\frac{dx}{dt}=-Y_{\frac{x}{s}}\frac{dS}{dt}$

c$.$ By comparing the form of logistic equation derived in part $($a$)$ with the Monod equation at low substrate concentrations $($first$-$order kinetics$),$ determine the logistic equation constant $(k)$ in terms of Monod kinetic constants.