MAT244-2013S > Ch 1--2

2.5 - Confusion about Population Dynamics Derivation

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Christopher Long:
In advance, I'm sorry that the way that the following math is done isn't gonna be super pretty, math typing isn't my thing.
Anyways, in Boyce & DiPrima 9 ed. pp. 82 (may be slightly different in 10 ed.), the assertion is made that

y' = ry(1-y/k) simplifies to y = (y0K)/(y0+((K-y0)e-rt)

In the middle of the derivation, int([1/K]/[1-y/K]) is presumed to equal -ln(1-y/K)

However,

int((1/K)/(1-y/K))
=int(1/(K-y))
=-ln(K-y)

Plugging this into WolframAlpha also yields the answer -ln(k-y)

What accounts for this discrepancy?

Victor Ivrii:
I got it (albeit with the difficulty since decrypting is not my thing :D)

There are two "different" answers  $-\ln (K-y)$ and $-\ln (1-y/K)=-\ln [(K-y)/K]= -\ln (K-y) -\ln K$.

The difference is a constant $-\ln K$ but we got them by integrating so the answer is defined modulo additive constant anyway.

Christopher Long:
That's sneaky. Very, very sneaky. Thanks for the clarification!