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Ch 1--2 / Re: 2.5 - Confusion about Population Dynamics Derivation
« on: January 30, 2013, 09:19:17 PM »
That's sneaky. Very, very sneaky. Thanks for the clarification!
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Ch 1--2 / 2.5 - Confusion about Population Dynamics Derivation
« on: January 30, 2013, 07:58:54 PM »
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?
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?
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