Consider the differential equation dy dx =y ,(2-y)

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

Problem

Consider the differential equation

\frac{dy}{dx}=y\,(2-y).

A student claims that this equation models the rate of change of a quantity

y

with respect to

x

. (a) Determine all equilibrium solutions. (b) For each equilibrium solution found in part (a), classify it as stable or unstable. Justify your classification by analyzing the sign of

\dfrac{dy}{dx}

on intervals separated by the equilibrium values. (c) Describe the long–term behavior of solutions that satisfy

y(0)=0.5

and

y(0)=3

x\to\infty

. Support your conclusions with clear reasoning based on your work in part (b).

Related theory

Qualitative Analysis
Applications
Differential Equations

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