Toronto Math Forum
MAT2442013S => MAT244 MathLectures => Ch 12 => Topic started by: Christopher Long 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(1y/k) simplifies to y = (y_{0}K)/(y_{0}+((Ky_{0})e^{rt})
In the middle of the derivation, int([1/K]/[1y/K]) is presumed to equal ln(1y/K)
However,
int((1/K)/(1y/K))
=int(1/(Ky))
=ln(Ky)
Plugging this into WolframAlpha also yields the answer ln(ky)
What accounts for this discrepancy?

I got it (albeit with the difficulty since decrypting is not my thing :D)
There are two "different" answers $\ln (Ky)$ and $\ln (1y/K)=\ln [(Ky)/K]= \ln (Ky) \ln K$.
The difference is a constant $\ln K$ but we got them by integrating so the answer is defined modulo additive constant anyway.

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