1. Autonomous Equations; Phase Lines
MATH 374 Lecture 9
Autonomous Equations; Phase Lines

2. 3.4: Autonomous Equations and Phase Lines
Definition: A first order differential equation of the form
is said to be autonomous.
Example 1: Some autonomous equations:

3. Behavior of Solutions of (1)
Since the RHS of (1) is a function of y alone, we can look at the sign of f(y) to determine where solutions of (1) are increasing, decreasing, or constant!
Also, using (1) and the Chain Rule, we find that if f is differentiable, then
Therefore, we can determine concavity of solutions of (1) from f(y)!

4. Example 2
Investigate the behavior of solutions of the logistic equation (3):
Solution:
Here, f(y) = by – ay2, with a>0 and b>0 assumed.
The function f is a parabola with zeros at y = 0, y = b/a, and vertex at y = b/2a.
We call the y-axis a phase line.
When y = 0 or y = b/a, y’ = 0.
We call y ´ 0 and y ´ b/a equilibrium solutions.
When y > b/a or y < 0, y’ < 0, ) y is decreasing.
When y 2 (0, b/a), y’ > ) y is increasing.

5. Example 2 (continued) Now look at the concavity of solutions via (5).
Here f’(y) = b – 2ay
) f’(y) = 0 if y = b/2a,
) f’(y) > 0 if y < b/2a,
) f’(y) < 0 if y > b/2a.
Note: We call y ´ 0 and unstable equilibrium solution and y ´ b/a an asymptotically stable solution.

6. Example 3
Look at the direction field for the logistic equation with b = 3 and a = 2.