A finite portion of the slope field for the differential equation is described below
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A finite portion of the slope field for the differential equation dtdy=f(y) is described below.
• Along the horizontal line y=−2, every segment of the slope field is horizontal.
• Along the horizontal line y=3, every segment of the slope field is horizontal.
• For points with −2<y<3, the field segments tilt downward from left to right.
• For points with y<−2 or y>3, the field segments tilt upward from left to right.
Answer the following questions about solution curves to this differential equation.
(a) Identify all equilibrium solution values of y. Explain how you arrived at your conclusion using the information given about the slope field.
(b) Classify each equilibrium solution you identified in part (a) as stable, unstable, or semi-stable. Justify your classifications by referring to the direction of nearby field segments.
(c) Consider the solution curve passing through the point (0,0). Describe the long-term behavior of this solution as t→∞ and as t→−∞. Provide reasoning that connects your description to the slope field information.