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Difficulty guide

The label describes the problem, not your ability.

1/5 - Foundational

Warm-up setup; checks one basic idea.

2/5 - Routine exam skill

Normal syllabus skill; usually one main rule or interpretation.

3/5 - Standard multi-step

Exam-style problem with multiple steps or a common trap.

4/5 - Hard exam skill

High-end exam problem; combines methods carefully.

5/5 - Challenge

Stretch problem beyond normal exam pressure.

Question

Consider the differential equation dydx=y(4−y).\frac{dy}{dx}=y(4-y).
Total: 5 points

(a) Find the equilibrium solutions of the differential equation. (1 point)

(b) For a solution with y(0)=1y(0)=1, determine whether the solution is increasing or decreasing at x=0x=0. (1 point)

(c) For 0<y<40<y<4, determine the value of yy where dy/dxdy/dx is greatest. (2 points)

(d) Describe the long-term behavior suggested by the differential equation for a solution with 0<y(0)<40<y(0)<4. (1 point)

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