Need Help with Integrating Equations for u and YX/S: Step-by-Step Guide

[staceybiomed]

Homework Statement

Using the following equations,
dX/dt = uX

Derive the following equation:
u₀*t = (K_S*Y_X/S + S₀*Y_X/S + X₀) * ln(X/X₀) - (K_S*Y_X/S)/(S₀*Y_X/S + X₀) * ln{(S₀*Y_X/S + X₀ - X)/(S₀*Y_X/S)}

Homework Equations

u =u₀*S/(K_S + S)
Y_X/S = (X - X₀)/(S₀ -S)

The Attempt at a Solution

Using the equation for YX/S, I solved for S and then plugged that and the equation for u into the equation for dx/dt. After some rearranging and use of common denominators, I have the following:

u₀*dt = (Y_X/S*K_S + Y_X/S*S₀ - X + X₀)/{X*(Y_X/S - X + X₀)} dx

Integration of the left side is easy and results in u₀*t but I'm still struggling with the left side.

Can anybody please help? My calculus is a little rusty. I have to be able to show my work so using a integration calculator won't work. Thanks!

 

[staceybiomed]

I see that numerous people have viewed my post but nobody has replied. Does this mean that the problem is as difficult as I think it is? :-(

 

[LCKurtz]

I'm guessing the problem is stated so poorly nobody can figure out what the problem actually is. You give a simple DE with X in terms of t and the answer contains all kinds of other variables -- K, Y, S and "relevant equations" that don't seem to have anything to do with the original.