Does someone have a solution for the following equation where the carrying capacity varies linearly with time (and only time):

$$\frac{\mathrm dN}{\mathrm dt}= rN\left(1-\frac{N}{K}\right)$$

$K = m \cdot t + b$ where $m$ and $b$ are constant.

I'm looking for a solution that models population growth with a linearly increasing upper limit:

population growth model

I'm looking for an equation that is in the form of $N$ as a function of $t$ where $r$, $m$, and $b$ are constants.

Sorry, the last math class I took was multivariate calculus in college, so I'm not even sure this is a valid question.


Riccati's method still works. Set $y(t) = \frac{1}{N(t)}$, then $y(t)$ satisfies $$\frac{d}{dt}y(t) = - N^{-2}(t)\frac{d}{dt}N(t) = - r N^{-1}(t) + rK^{-1}(t) = -r y(t) + rK^{-1}(t) \, . $$
So $w(t) = e^{rt}y(t)$ satisfies $$ w'(t) = re^{rt} K(t) = \frac{re^{rt}}{mt + b} \, . $$ This leads to an integral of the form $\int s^{-1}e^s \, ds$ which does not have a solution in elementary functions (I think).


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