the relative growth rate of a population is proportional to

[math]

the relative growth rate of a population is proportional to the sum of the population and its reciprical. if the population is zero when t= pi/3. find P(t).

Where do I start? Thanks

 

[skeeter]

"the relative growth rate of a population is proportional to the sum of the population and its reciprocal."

\(\displaystyle \L \frac{dP}{dt} = k\left(P + \frac{1}{P}\right)\)

k is the constant of proportionality

\(\displaystyle \L \frac{dP}{dt} = k\left(\frac{P^2 + 1}{P}\right)\)

\(\displaystyle \L \frac{P}{P^2 + 1} dP = k dt\)

take it from here?

 

[stapel]

Where do I start?

Probably the best place to start would be with the problem statement:

Define a variable.
Create an expression for "its reciprocal".
Create an expression for "the sum of" the variable and the reciprocal.
Create the proportionality equation.
Use the given data point and see what you end up with.

Eliz.

 

[math]

thanks!

integral of P/(P^2 + 1) dp = kt + c
u = p^2 + 1 dx=du/2p
integral of p/u * du/2p
ln (P^2 + 1) / 2 = kt+ c
e^ (2kt + 2c) = P^2 + 1
p= sqrt ((ce^(2kt)) -1)
plug in 0 for p and pi/3 for t, but then i stilll have two variables to solve for: c and k. Did i do something wrong?

thanks again

 

[skeeter]

looks like you need one more data point for P, or some additional info regarding the rate of growth.

P = 0 at t = pi/3 ... ??? strange initial condition for a population problem.

recheck the original problem statement.