dP/dt=rP(1-P/k),
∫dP/(P(1-P/k))=∫rdt,
1/(P(1-P/k))=A/P+B/(1-P/k) where A and B are constants to be found.
A(1-P/k)+BP=1,
A-AP/k+BP=1.
Matching coefficients: A=1; -1/k+B=0, B=1/k.
1/(P(1-P/k))=1/P+1/(k-P).
∫(1/P+1/(k-P))dP=rt+C, where C is a constant.
ln(P)-ln|k-P|=rt+C,
ln|P/(k-P)|=rt+C.
If k<P, then P/(P-k)=ert+C, which can also be written: aP/(P-k)=ert, where a is a constant (C=-ln(a), a=e-C. If C=0, a=1).
aP=ert(P-k)=Pert-kert,
Pert-aP=kert,
P=kert/(ert-a).
Or P=k/(1-ae-rt).