Two slope fields are superimposed on the same -plane
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
Two slope fields are superimposed on the same xy-plane. Both arise from differential equations of the form dy/dx=a−y, but with different parameter values:
• Field A (dashed line segments): a=1
• Field B (solid line segments): a=3
Line segments are plotted at every lattice point having integer coordinates −2≤x≤2 and −1≤y≤5.
(a) Identify the equilibrium solution for each field.
(b) Describe how the entire family of solution curves for a=3 is related, in a vertical sense, to the family for a=1.
(c) Using the combined slope field, explain the long-term behaviour (as x→∞) of the solution that satisfies y(0)=0 when a=3.