
Differential Equations Ordinary and Partial Differential Equations Math Forum 
 LinkBack  Thread Tools  Display Modes 
November 20th, 2017, 04:43 AM  #1 
Newbie Joined: Jan 2014 Posts: 19 Thanks: 0  Population Model
I want to show that $p=10000\exp(\frac{t}{3}\ln\frac{23}{20})$ is the unique solution to the population growth problem ($t$ is in years): $\displaystyle dp/dt=kp, p(0)=10000, p(3)=11500$ In this case $\displaystyle t$ is restricted to nonnegative values (because $t$ is in years) and this restriction made it difficult to show that the solution is the unique solution. How to explain in words the solution is the unique solution by using the following existence and uniqueness theorem? Theorem: Let the functions $\displaystyle f$ and $\displaystyle âˆ‚f /âˆ‚p$ be continuous in some rectangle $\displaystyle Î± < t < Î²$, $\displaystyle Î³ < p < Î´$ containing the point $\displaystyle (t_0, p_0)$. Then, in some interval $\displaystyle t_0 âˆ’ h < t < t_0 + h, h>0$, contained in $\displaystyle Î± < t < Î²$, there is a unique solution $\displaystyle p = Ï†(t)$ of the initial value problem $\displaystyle p' = f (t, p), y(t_0) = p_0$. Last edited by woo; November 20th, 2017 at 04:49 AM. 
November 20th, 2017, 08:56 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,887 Thanks: 1836 
By using $e^{kt}$ as an integrating factor, the equation's solution is $p = 10000e^{kt}$ in order that $p(0) = 10000$. In order that $p(3) = 11500$, the value of $k$ must be $\frac13\ln\left(\frac{23}{20}\right)$. 
November 20th, 2017, 09:19 AM  #3  
Newbie Joined: Jan 2014 Posts: 19 Thanks: 0  Quote:
I want to show that the solution is a UNIQUE solution.  
November 20th, 2017, 02:57 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 19,887 Thanks: 1836 
If integrating the equation (after applying the integrating factor) leads to only one solution, that solution must be unique.


Tags 
model, population 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Population Modellogistic nonlinear firstorder ordinary differential equation  Jaider  Applied Math  0  April 10th, 2015 06:13 PM 
population changes  gonzo  Algebra  2  May 16th, 2013 09:00 PM 
differential equations  population model  mbradar2  Differential Equations  8  September 25th, 2010 02:53 PM 
Fixed points of a model of population  Seng Peter Thao  Applied Math  0  June 30th, 2007 11:53 AM 
Population Mean  symmetry  Advanced Statistics  0  April 7th, 2007 04:03 AM 