I have the following Logistic model equation (left out the values for the constants for simplicity), which I'm unable to solve for $P(t)$.

$\dfrac {dP} {dt} = kP \left (1- \dfrac P {P_\infty} \right)-H$

If the harvesting constant, $H$ was not present, then the ODE could be solved by Bernoulli's equation. The problem is that I am not sure how to start solving this specific type of differential equation.

Can someone point me in the right direction?

Thank you for your time!


The equation is in separated variables: $$ \frac{dP}{k\,P \left (1- \dfrac P {P_\infty} \right)-H}=dt. $$

  • $\begingroup$ Ah I see. But that looks like a beast to solve right? What would be your first step if you went this route? Thanks for the reply! $\endgroup$ – KidMe Mar 10 '15 at 21:24
  • $\begingroup$ It is polynomial in the denominator of degree two. Parcial fraction descomposition. $\endgroup$ – Julián Aguirre Mar 10 '15 at 23:55
  • $\begingroup$ Partial fractions? How would you start doing that, since you don't have 2 factors in the denominator. Thanks for the reply! $\endgroup$ – KidMe Mar 11 '15 at 12:14
  • $\begingroup$ You are supposed to find the factors. $\endgroup$ – Julián Aguirre Mar 11 '15 at 12:29
  • $\begingroup$ Thank you Julián for the help. I had managed to work it out in the end! $\endgroup$ – KidMe Mar 18 '15 at 17:59

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