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

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:

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)!

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’ > 0 ) y is increasing.

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.

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