differential equation, i dont get book's answer.
question: dP(t)/dt = kP(t) ( 1 - P(t)/M) , where P is population at time t, M is carrying capacity.books solution: P(t) = MCe^(kt)/ ( M + Ce^kt).
ok here is what i got.
first 1 - P/M = (M- P ) / M, so dP/dt = kP( M-P)/M , which is seperable autonomous.
we have integral M / [ P (M-P)] dP = integral k dt. using partial fractions i get
integral 1 / P + 1 / ( M-P) = kt + C
ln P - ln M-P = kt + C
ln | P / (M-P) | = kt + C
P /( M-P) = e^(kt + c)
P = (M-P) Ce^kt , where C = e^c
P = M Ce^kt -P*Ce^kt
P + P Ce^kt = M Ce^kt
P ( 1 + Ce^kt) = MCe^kt
P

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ok ill try to work it out and see if i get anything different

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