Chapter 7.5, Problem 40E

To determine

To explain whether statement is true or false and justify your answer.

Answer to Problem 40E

The given statement

The graph of the solution to the differential equation $dP/dt=kP\left(100-P\right)$ has asymptotes $y=0$ and $y=100$ is true.

Explanation of Solution

Given:

If $0<P\left(0\right)<M$ , then the graph of the solution to the differential equation $dP/dt=kP\left(100-P\right)$ has asymptotes $y=0$ and $y=100$ .

Calculation:

The differential equation is $dP/dt=kP\left(100-P\right)$

So, carrying capacity is $M=100$

Below given a logistic curve

  

From the graph of logistic curve , it is clear that there exist asymptote at $y=0$ and carrying capacity

Hence, the given statement

The graph of the solution to the differential equation $dP/dt=kP\left(100-P\right)$ has asymptotes $y=0$ and $y=100$ is true.