For the one-parameter family of differential equations dy dt =y ,(k-y)

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

For the one-parameter family of differential equations

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

k

is a non-zero constant parameter and

y

is a differentiable function of

t

. (a) In terms of

k

, determine all equilibrium solution values of

y

. (b) Suppose

k>0

. Classify each equilibrium solution as stable, unstable, or semi-stable. Justify your classifications by discussing the sign of

\dfrac{dy}{dt}

on open intervals between the equilibrium values. (c) Suppose

k<0

. Repeat the classifications in part (b) for this case, justifying your answers. (d) Describe how both the number of equilibrium solutions and their stability change as

k

passes through

0

. Explain the significance of your result in terms of long-term behavior of solutions.

Related theory

Qualitative Analysis
Applications
Differential Equations

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