**Example:** Let *P*(*t*) be the population of a certain animal species.
Assume that *P*(*t*) satisfies the logistic growth equation

**1.**- Is the above differential equation separable?
**2.**- Is the differential equation autonomous?
**3.**- Is the differential equation linear?
**4.**- Without solving the differential equation, give a sketch of
the graph of
*P*(*t*). **5.**- What is the long-term behavior of the population P(t)?
**6.**- Show that the solution is of the form
Find

*A*and*B*.

Hint: Use the initial condition and the result of 5. **7.**- Where is the solution's inflection point?

Hint: This can be done without using the answer of 6. **8.**- What is special about the growth rate of the population
*P*(*t*) at the inflection point (found in 7)?

**1.**- This differential equation is autonomous, i.e. the variable
*t*is missing. Therefore, this equation is indeed separable. **2.**- The answer is Yes! (see 1.) Every autonomous differential equation is separable.
**3.**- The equation is not linear because of the presence of .
**4.**- The graph of the solution
*P*(*t*) is as follows:

**5.**- Clearly, because of the initial condition (see the graph of the solution below), we have
,

200 being the carrying capacity.

**6.**- Let us solve this equation (use the technique for solving separable equations). First, we look for the constant solutions (equilibrium points or critical points). We have
Then, the non-constant solutions can be generated by separating the variables

,

and the integration

.

Next, the left hand-side can be handled by using the technique of integration of rational functions. We get

,

which gives

Hence, we have

.

Easy algebraic manipulations give

,

where .

Therefore, all the solutions arewhere

*C*is a constant parameter.

**Remark:**We may rewrite the non-constant solutions as,

where

*a*and*B*are two parameters. If we use the conditionswe will be able to get the desired solution. Indeed, we have

Thus, and consequently .

**7.**- From the graph (see 4.), the graph of
*P*(*t*) has an inflection point between 0 and 200. Let us find it. Since we must have*P*''(*t*) = 0, we get,

where we used the chain rule and the fact that . Since our solution is not one of the two constant solutions we are only left with the equation

This simplifies to

,

which gives

*P*=100 (half-way between 0 and 200).

**Remark:**You still need to convince yourself that*t*=100 is indeed the inflection point, that is the second derivative changes sign at that point. **8.**- The growth rate at
*t*=100 is maximal.

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