Calculus for enzyme kinetics I will try my best to make sense.   Calculus is a distant memory for me...
I would like to see if there is a way to generate an equation to determine the amount of remaining substrate as a function of time in an ezymatic reaction using parameters for Vm, Km and C0, and with a twist- the amount of water will decrease at a particular rate.
So, without the water decrease complication, the basic idea is that the substrate concentration decreases as 
dC/dt  =  - (VmC/(Km + C) )
Vm and Km are parameters.
So, with a C0 (starting concentation) a Vm and Km and a t, I would like to be able to determine C.
If that can be solved that would be a good first step.
But, the next step is as follows.
Actually   dC/dt = - ( VmC/(Km+C))/V
where V is volume.
But volume is changing by a particular percentage such that
dV/dt = -(A * V)
So, I would ultimately like to solve for C given Vm, Km, Co, A, V0 and time.
I've figured out how to get answers to this by brute force using Microsoft Excel with small slices of time, but if there is a way to do it more elegantly that would be preferable.   Or, if I can get an answer that there is no simple elegant way, that would be helpful, too.
Thanks
 A: First solve the "exponential decay" equation for $V$ explicitly, or recall the general solution.  We get $V=V_0e^{-At}$. Then rewrite the original equation as 
$$\frac{K_m+C}{V_m C}\frac{dC}{dt}=-\frac{1}{V_0}e^{At}$$
and integrate. On the left, forgetting about the constant of integration, we get $\frac{K_m}{V_m}\ln C +\frac{C}{V_m}$. On the right we get $-\frac{1}{V_0 A}e^{At}+D$, where $D$ is the constant of integration. By setting $t=0$, we can express $D$ in terms of our parameters and the initial value $C_0$. 
We end up with an equation that links $C$ and $t$. However, it is an implicit equation. One can solve explicitly for $t$ in terms of $C$ using elementary functions. However, one cannot solve for $C$ explicitly in terms of $t$ using elementary functions. 
A: Assuming I understood the problem correctly(I have no knowledge in biochemistry or whatever the subject you're studying is), the answer to the first DE is $$C= K_m  W\left (e^{\frac{\frac{C_0}{K_m}-\frac{V_m t}{K_m}}{K_m}}\right )$$
I'm working on the second.
I'm also assuming that giving the solution is pointless, but if anyone asks for it I'll be glad to TeX it.
The answer to the second one is $$C=\frac{V_0(V_m C_0-K_m)}{V_m e^{A t}+V_0}$$
