Consider the family of linear differential equations dy dx =c ,y+2

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Consider the family of linear differential equations

\frac{dy}{dx}=c\,y+2,

where the constant

c

is a real-valued parameter. (a) Identify the

y

-value at which slope field segments are horizontal for an arbitrary

c

and explain how this horizontal line changes as

c

varies. (b) Describe qualitatively how the density and direction of the slope field line elements depend on the sign and magnitude of

c

. (c) Sketch slope fields for

c=-2

c=0

, and

c=1

on separate, clearly labeled axes of identical scale. Emphasize similarities and differences among the three fields. (d) Explain how the slope field for

c=0

serves as a transitional case between the fields for negative and positive

c

. Include a discussion of solution concavity and long-term behavior in your answer.

Related theory

Qualitative Analysis
Visualization
Differential Equations

Browse the problem library