A first-order autonomous differential equation is represented by the following verbal…
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A first-order autonomous differential equation y′=f(y) is represented by the following verbal description of its slope field.
• For every point in the xy-plane, the slope depends only on the y-coordinate.
• Horizontal line segments (slope 0) appear exactly along the line y=2.
• Above the line y=2 the slope field shows positive slopes that become steeper as y increases; below y=2 the slopes are negative and become more negative as y decreases.
• The field suggests that all solution curves are vertical translations of one another.
Use this information to analyze the differential equation and its solutions.
(a) Describe the relationship between the family of solution curves and the structure of the slope field given above.
(b) Explain how specifying the initial condition y(0)=1 determines a unique solution curve. In your explanation, refer explicitly to features of the slope field.
(c) Suppose no initial condition is provided. Describe the set of possible solution curves and explain in what sense they differ from one another.