#$&*
course MTH 279
Query 08 Differential Equations*********************************************
Question: 3.5.6. Solve the equation dP/dt = r ( 1 - P / P_c) P + M with r = 1, P_c = 1 and M = -1/4.
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
Plugging in our values we have the equation:
dP/dt = 1 (1 - P / 1) P - 1/4. Simplifying allows us to see our equation as dPdt = (1 - P / 1) P - 1/4 which simplifys further to be: dP/dt = P - (P^2)/P - 1/4, which equals dPdt = -1/4.
confidence rating #$&*:
@&
If r = 1, P_c = 1 and M = -1/4, our equation becomes
dP/dt = (1 - P) * P + M,
which we can rewrite as
dP / ( -P^2 + P - 1/4) = dt.
The denominator factors into -(P - 1/2) ^ 2 so we have
-dP / (P - 1/2)^2 = dt,
which is easily integrated to obtain
1 / (P - 1/2) = t + c
so that
P = 1 / (t + c) + 1/2.
At t -> infinity, P approaches 1/2, which is half the 'carrying capacity' P_c = 1 of the system.
*@
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution:
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary): Something is wrong here..............
------------------------------------------------
Self-critique rating:
3.5.10. Solve dP/dt = k ( N - P) * P with P(0) = 100 000 assuming that P is the number of people, out of a population of 500 000, with a disease.
Assume that k is not constant, as in the standard logistic model, but that k = 2 e^(-t) - 1. Plot your solution curve and estimate the maximum value of P, and also that value of t when P = 50 000.
Interpret all your results in terms of the given situation.
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
dP/dt = 2 e^(-t) - 1 (500000 - P) * P simplifies to dP/dt = (2 e^(-t) - 1)500000P - P^2, which can be put into form dP/dt = P^2 - 500000(2 e^(-t) - 1)P.
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution:
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary): I'm so confused here and I don't see how this relates to chapter 3 section 5.....
------------------------------------------------
Self-critique rating:"
@&
I've included a soltution to the first problem. See if you can get the second.
These equations are separable, which should give you some comfort.
The method of partial fractions is commonly used to solve these equations. That wasn't necessary in the first problem, since the constants resulted in a perfect square, but in most cases you will use partial fractions.
*@
@&
&#Please see my notes and submit a copy of this document with revisions, comments and/or questions, and mark your insertions with &&&& (please mark each insertion at the beginning and at the end).
Be sure to include the entire document, including my notes. &#
*@