My Math Forum  

Go Back   My Math Forum > College Math Forum > Differential Equations

Differential Equations Ordinary and Partial Differential Equations Math Forum


Reply
 
LinkBack Thread Tools Display Modes
November 20th, 2017, 04:43 AM   #1
woo
Newbie
 
Joined: Jan 2014

Posts: 14
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 $\displaystyle ∂f /∂p$ be continuous in some rectangle $\displaystyle α < t < β$, $\displaystyle γ < p < δ$ containing the point $\displaystyle (t_0, p_0)$. Then, in some interval $\displaystyle t_0 − h < t < t_0 + h, h>0$, contained in $\displaystyle α < t < β$, there is a unique solution $\displaystyle p = φ(t)$ 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.
woo is offline  
 
November 20th, 2017, 08:56 AM   #2
Global Moderator
 
Joined: Dec 2006

Posts: 18,241
Thanks: 1438

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 $\frac13\ln\left(\frac{23}{20}\right)$.
skipjack is offline  
November 20th, 2017, 09:19 AM   #3
woo
Newbie
 
Joined: Jan 2014

Posts: 14
Thanks: 0

Quote:
Originally Posted by skipjack View Post
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 $\frac13\ln\left(\frac{23}{20}\right)$.
@skipjack

I want to show that the solution is a UNIQUE solution.
woo is offline  
November 20th, 2017, 02:57 PM   #4
Global Moderator
 
Joined: Dec 2006

Posts: 18,241
Thanks: 1438

If integrating the equation (after applying the integrating factor) leads to only one solution, that solution must be unique.
skipjack is offline  
Reply

  My Math Forum > College Math Forum > Differential Equations

Tags
model, population



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Population Model-logistic nonlinear first-order 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





Copyright © 2017 My Math Forum. All rights reserved.