differential equation help,

$\displaystyle \frac{dS}{dt} = \frac{K_{cat}E_{0}S}{K_{m} + S} $

$\displaystyle \frac{dP}{dt} = \frac{K_{cat}E_{0}S}{K_{m}+S} $

with intial conditions $\displaystyle S(0) = S_{0} $ $\displaystyle P(0) = p_{0) $

show that S and P satisfy a conservation law given by S+P = c,

for some constant c, and find the value of c. Hence determine $\displaystyle \lim_{t\ \infty} P(t) $

I know that if you add the two equations you get a 0, is that what c equals?

Re: differential equation help,

From $\displaystyle \frac{dS}{dt}= \frac{K_{cat}E_0S}{K_mS}$ and $\displaystyle \frac{dP}{dt}= \frac{K_{cat}E_0S}{K_mS}$ subtracting, $\displaystyle \frac{dS}{dt}+ \frac{dP}{dt}= 0$.

We also have S+ P= c for constant c, differentiating both sides, we have $\displaystyle \frac{dS}{dt}- \frac{dP}{dt}= 0$.

So subtracting the second equation from the first, we have $\displaystyle 2\frac{dP}{dt}= 0$ from which it follows that P(t) is a constant.

Re: differential equation help,

Sorry I have written out the equation wrong, I have now corrected it, also I dont understand how you got a minus, after differentiating both sides s+p = c.?

Re: differential equation help,

Quote:

Originally Posted by

**Tweety** $\displaystyle \frac{dS}{dt} =- \frac{K_{cat}E_{0}S}{K_{m} + S} $

$\displaystyle \frac{dP}{dt} = \frac{K_{cat}E_{0}S}{K_{m}+S} $

with intial conditions $\displaystyle S(0) = S_{0} $ $\displaystyle P(0) = p_{0) $

show that S and P satisfy a conservation law given by S+P = c,

for some constant c, and find the value of c. Hence determine $\displaystyle \lim_{t\ \infty} P(t) $

I know that if you add the two equations you get a 0, is that what c equals?

The first equation should have a minus infront of it