The Logistic Equation (from a video)

[smutangama]

Homework Statement

I can't really post every single equation he used but I still need the help.

I can follow him fine up until about 03:15 where he says, "I'm going to pick clever values for P"

Are you allowed to do that? How do you know what "clever values" to use and how do you apply them?

Homework Equations

The Logistic Equation and the Analytic Solution - YouTube

The Attempt at a Solution

no attempt, just need explanation.

Thanks.

 

[CAF123]

He is just doing a partial fraction decomposition writing K/P(K-P) = A/P + B/(K-P). The resulting eqn K = A(K-P) + BP holds for all P, so he conveniently chose P=0 so as to eliminate the second term on the RHS and thereby obtain a value for A. Similarly, choosing P=K allows cancellation of the first term on the RHS and so a value for B can then be found easily.

There are other methods to find A and B, though.

 

[smutangama]

He is just doing a partial fraction decomposition writing K/P(K-P) = A/P + B/(K-P). The resulting eqn K = A(K-P) + BP holds for all P, so he conveniently chose P=0 so as to eliminate the second term on the RHS and thereby obtain a value for A. Similarly, choosing P=K allows cancellation of the first term on the RHS and so a value for B can then be found easily.

There are other methods to find A and B, though.

What other methods can I use? Can you please explain them? Thanks

 

[CAF123]

What other methods can I use? Can you please explain them? Thanks

Instead of plugging in convenient values of P, from the eqn K = A(K-P) + BP, write K = (B-A)P + AK.

Now compare LHS and RHS. It implies the coefficient of P is zero and that A is 1. Is it clear?
Then B-A = 0 and since A=1, B is then also 1.

 

[smutangama]

Thanks! i just hit myself in the head for not noticing that. I guess stress can really screw with your mind. Thanks again.