A slope field for an autonomous differential equation exhibits the following…
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A slope field for an autonomous differential equation dxdy=f(y) exhibits the following qualitative features.
• Horizontal line segments (zero slope) occur precisely at y=−2 and y=3.
• For −2<y<3, the short line segments tilt upward from left to right (positive slope).
• For y>3, the short line segments tilt downward from left to right (negative slope).
• For y<−2, the short line segments again tilt upward from left to right (positive slope).
Assume the slope field has no other critical features.
(a) Identify all equilibrium solutions.
(b) Classify each equilibrium solution as stable or unstable, citing specific evidence from the described arrow directions.
(c) Using your classifications, describe the behavior of a solution curve that satisfies y(0)=0. In particular, state the limit of y as x→∞ and justify your answer.