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I have the following problem: assuming the growth of a certain population is described by

$\displaystyle{\frac{dN}{dt}=rN(1-\frac{N}{K})-EN}$,

is there a way to find a solution to this differential equation analytically. I am thinking seperation of variables,

$\displaystyle {\int \frac{1}{rN(1-\frac{N}{K})-EN}dN=\int 1 dt}$.

But I am not sure how to evaluate the left integral.

matte_studenten
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    This has been asked many times before... For example, see this question and the linked questions there. Anyway, you can also change to $y(t)=1/N(t)$ (to find all nonzero solutions; $N(t)=0$ is trivially a solution). – Hans Lundmark Apr 12 '21 at 17:12

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Multiply top and bottom by $K$, and factor $N$ out of the denominator. The denominator is now $N(rK-rN-EK)$ which you can break up with partial fractions. I pull out $\frac{-K}{r}$ and define $C = K+\frac{EK}{r}$ to get $$\frac{-K}{r}\cdot \frac{1}{N(N-C)}$$ which you can break up with partial fractions and integrate. I get $$\frac{A}{N} + \frac{B}{N-C}$$ where $A = \frac{-1}{C}$ and $B = 1 + 1/C$.

RobertTheTutor
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