- #1
gpax42
- 25
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Hi all, it's been a over a year since I took my differential equations and linear algebra course and I'm currently enrolled in a class that assigned this problem as a sort of refresher on analytically solving differential equations. I can't seem to remember the proper approach to going about all this and I'm no longer in possession of my textbook from the previous course. Any help would be greatly appreciated ![Red Face :redface: :redface:](https://www.physicsforums.com/styles/physicsforums/xenforo/smilies/oldschool/redface.gif)
Homework Statement
Using methods for solving differential equations exactly, solve the Verhulst equation for logistic population growth
[tex]\frac{dP}{dt}[/tex]= rP(1 - [tex]P/K[/tex])
where r is the growth rate and K is the carrying capacity.
The attempt at a solution
I began by altering the equation so as to create a more solvable form...
Dividing through by K on both sides gives me
[tex]\frac{d}{dt}[/tex][tex]\frac{P}{K}[/tex] = r[tex]\frac{P}{K}[/tex](1 - [tex]P/K[/tex])
I then set x = [tex]P/K[/tex]
[tex]\frac{dx}{dt}[/tex] = rx(1-x)
at this point I know I need to integrate both sides
dx = rx(1-x)dt
following integration...
x(t) = [rx(1-x)]t + C
is this even remotely correct?
Thanks again for any help you can lend me
-gpax42
![Red Face :redface: :redface:](https://www.physicsforums.com/styles/physicsforums/xenforo/smilies/oldschool/redface.gif)
Homework Statement
Using methods for solving differential equations exactly, solve the Verhulst equation for logistic population growth
[tex]\frac{dP}{dt}[/tex]= rP(1 - [tex]P/K[/tex])
where r is the growth rate and K is the carrying capacity.
The attempt at a solution
I began by altering the equation so as to create a more solvable form...
Dividing through by K on both sides gives me
[tex]\frac{d}{dt}[/tex][tex]\frac{P}{K}[/tex] = r[tex]\frac{P}{K}[/tex](1 - [tex]P/K[/tex])
I then set x = [tex]P/K[/tex]
[tex]\frac{dx}{dt}[/tex] = rx(1-x)
at this point I know I need to integrate both sides
dx = rx(1-x)dt
following integration...
x(t) = [rx(1-x)]t + C
is this even remotely correct?
Thanks again for any help you can lend me
-gpax42