Question

Which of the following statements is true for the logistic differential equation?

The graph has a horizontal asymptote at y = 18.
y is growing the fastest when y = 9.
The limiting value for y is 18.
All of the above.

Answer 1

All of the above

Explanation:

dy/dt = y/3 (18 − y)

0 = y/3 (18 − y)

y = 0 or 18

d²y/dt² = y/3 (-dy/dt) + (1/3 dy/dt) (18 − y)

d²y/dt² = dy/dt (-y/3 + 6 − y/3)

d²y/dt² = dy/dt (6 − 2y/3)

d²y/dt² = y/3 (18 − y) (6 − 2y/3)

0 = y/3 (18 − y) (6 − 2y/3)

y = 0, 9, 18

y" = 0 at y = 9 and changes signs from + to -, so y' is a maximum at y = 9.

y' and y" = 0 at y = 0 and y = 18, so those are both asymptotes / limiting values.