Solving the logistic growth model

[jwang34]

The logistic growth model is the following:

dx/dt=rx(1-x/K), with r and K and as constants, and x is a function of t.

I'm really not sure where to begin. First I tried separation of variables, but that didn't work out (and I don't even know if I was doing it right). Should I even be looking for an integration factor in solving this? It looks simple...but I guess I'm rusty in this.

The end result is supposed to be:

x(t)=K/(1+ce^-rt) c=[K-x(0)]/x(0)

Second, I tried deriving this equation and getting it to look like the previous equation, but I think I'm missing somethings.

So just a tip or hint that can push me down the right track would be great. Thanks a lot!

 

[Dick]

Separation of variables is the right way to go. It gives you an x integral you can do easily by partial fractions. The rest is algebra. Get started and if you get stuck let us know.

 

[jwang34]

So separation of variables in this case would be A(t)dt+B(x)dx=0. So I would have dx/rx=(1-x/K)dt. Then, I should integrate both sides? Is this the right track?

 

[Dick]

Put ALL of the x's on one side with the dx.

 

[jwang34]

So I get dx/(rx(1-(x/K))=dt. Then I should use partial fractions to integrate?

 

[Dick]

Exactly. Use partial fractions.