DE equation modeling growth off a tan function

[Outlaw747]

Homework Statement

Data suggests rate of growth of bacteria is proportional to the tangent function evaluated on the amount of bacteria present at time t, up to 1.5 mg. After 1.5 mg, value maintained. pi/6 to pi/3 mg growth in 1 hour. How would I write a DE equation based on this? I'm not asking how to solve it but to set the equation up.

Homework Equations

dP/dt = kP
P(t) = P-initial * e^(kt)

The Attempt at a Solution

I am not really sure how to go about this. If we use the second equation isn't that based off another growth model? If I have dP/dt = tan (t) , that doesn't really make sense. Just need a push in the right direction if possible. Tried a few different attempts but none really worked.

 

[Gib Z]

The question stated "is proportional to", not "equals", so that should be dP/dt = k tan(t). That's only until 1.5mg though. If I understand correctly, after 1.5 mg, P increases by at least pi/6 mg, and at most pi/3 mg, each hour? For that part I would suggest something like a piecewise definition- Before 1.5mg, dP/dt = k tan(t), after 1.5mg, $$\pi/6 < dP/dt < \pi/3$$.

 

[Outlaw747]

K I'll try that, thanks.