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.
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 $B = 1 + 1/C$.