# Math Help - A simple ODE problem

1. ## A simple ODE problem

Hi everyone,

I have a problem understanding an ODE and using it to find something particular. Consider the following :

ODE : dC/dt= S-r*C
where

S: synthesis rate
r : death rate
C: population
Co: initial population

the analystical solution is simply C(t) =S/r -(S/r-Co)*exp(-r*t)

My questions are : i) How can we calculate the population that has died at time t
ii) How can we calculate the population that has synthesized at time t
My problem in understanding is relevant the rates S and r and how these can be applied to evaluate the exact quantity of population that has followed death and generation.
So If I want to find how many have died (or synthesized) at time t (given r or S) is it simply the multiplication with C(t) or with C(t-1)?

I would appreciate your help with this problem. Thanks a lot.

2. ## Re: A simple ODE problem

Originally Posted by Apostolos1234
Hi everyone,

I have a problem understanding an ODE and using it to find something particular. Consider the following :

ODE : dC/dt= S-r*C
where

S: synthesis rate
r : death rate
C: population
Co: initial population

the analystical solution is simply C(t) =S/r -(S/r-Co)*exp(-r*t)

My questions are : i) How can we calculate the population that has died at time t
ii) How can we calculate the population that has synthesized at time t
My problem in understanding is relevant the rates S and r and how these can be applied to evaluate the exact quantity of population that has followed death and generation.
So If I want to find how many have died (or synthesized) at time t (given r or S) is it simply the multiplication with C(t) or with C(t-1)?

I would appreciate your help with this problem. Thanks a lot.
Wouldn't the amount of the population that has died be equal to the difference between the starting population and the current population?

3. ## Re: A simple ODE problem

I suppose that S= so and so deaths per unit time
and r= so and so deaths per unit time.
So if you want to find how many took birth and how many got fried, you just have to plug in the relevant valur for t in this ?

4. ## Re: A simple ODE problem

Concerning the first repply,

I am sorry but I do not get your point. There are two parallel processes taking place. Death and generation with rates r and S respectively.
Lets say at time 0 these rates are 0 and for all different times t remain constant at values greater than zero. If I would substract Co from C(t) what do I find ?

For the second repply again I can get your point.
The analytical solution displays the population at any time t given the rates S and r. These rates are positive values and zero only at time zero. I dont understand....Where to plug the "relevant value of t" and what how to find that?

Thank you.

Apostolos

5. ## Re: A simple ODE problem

I am wondering....considering the previous model.

Is it true that ??:

C(current time)=

C(prev.time) + S(current time) - r*[ C(prev.time) + S(current time)]

where C : population
S : newly synthesised polulation (assume that is independt of population size(const.))
r : death rate of population (assume constant)
and if also the population that dies at any time t is labeled with D we must have :

D(current time)=r*[ C(prev.time) + S(current time)] ??

Please anyone who can take a look at it and let me know an opinion about it.

Thanks!