The rate of change of a certain population is proportional to the square root of its size. Model this situation with a differential equation.

The solution is said to be $\dfrac{dP}{dt} = k\sqrt{P}$, where $k > 0$ is the proportionality constant.

Since the problem statement says rate of change instead of rate of increase, isn't it true that $k$ can be both positive and negative rather than just positive?

I would greatly appreciate it if people could please take the time to clarify this.

  • 2
    $\begingroup$ I agree that $k$ can be either positive or negative. $\endgroup$ – callculus Jun 24 '17 at 17:38
  • $\begingroup$ I would even argue that "rate of increase" can be negative. To force the rate of change to be positive, you can either say so explicitly ("the rate of change is positive") or you can imply it with other words such as, "A population is growing with a rate of change that is ... ." $\endgroup$ – David K Jun 24 '17 at 18:13
  • $\begingroup$ The "rate" of change could also be interpreted to be the change relative to the value, $\dot P/P$ so that the ODE becomes $\dot P=k·P^{3/2}$. $\endgroup$ – LutzL Jun 24 '17 at 19:01

It certainly could be negative. The person who wrote the solution probably mean $|k| > 0$ instead, because if $ k = 0 $, then there is no growth at all. But a negative k is perfectly valid.


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