The function satisfies the differential equation dy dt =y ,(4-y)

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

Problem

The function

y(t)

satisfies the differential equation

\frac{dy}{dt}=y\,(4-y),

where

t

is measured in minutes and

y

in milligrams. A student begins an experiment with

y(0)=1

. (a) Determine all equilibrium solution values of

y

for this differential equation. (b) For each equilibrium value found in part (a), classify it as stable, unstable, or semi-stable. Support your answer by analyzing the sign of

\dfrac{dy}{dt}

above and below each equilibrium value. (c) Using your work in part (b), determine

\displaystyle\lim_{t\to\infty} y(t)

for the given initial condition. Explain your reasoning. (d) On axes that you provide, sketch a qualitative graph of

y(t)

for

t\ge 0

that is consistent with parts (a)–(c). Label all equilibrium values and the initial condition.

Related theory

Qualitative Analysis
Applications
Differential Equations

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