# differential equation help,

• May 19th 2013, 11:19 AM
Tweety
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?
• May 19th 2013, 02:39 PM
HallsofIvy
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.
• May 20th 2013, 04:12 AM
Tweety
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.?
• May 20th 2013, 04:15 AM
Tweety
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