Problem 1
Equilibria are 0 and 2; 0 is unstable for nearby positive states, and 2 is asymptotically stable.
Faculty use
Autonomous Equations, Equilibria and Stability
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Negative, so solutions move left/down toward y=1.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind using sign charts and phase lines to classify long-term behavior. Use the formulas y′=f(y); equilibrium y* satisfies f(y*)=0.
Problem 4
A strong error analysis identifies the first step that violates classification, algebra, theorem hypotheses, units, initial/boundary data, or numerical assumptions; then repairs that step and re-verifies the final result.
Problem 5
Application responses vary. Require defined variables, a compatible differential-equation model connected to motion, heat flow, circuits, fluids, population change, and engineering balance laws, a complete computation, interpretation, and limitations.
Problem 6
Accept two independent checks appropriate to the task: differentiation/substitution, initial or boundary conditions, dimensional analysis, phase-line or stability evidence, residuals, step refinement, conservation/energy checks, data comparison, or a second method.
Scoring guidance
Six problems × 10 points: 5 mathematical accuracy, 2 structure/justification, 2 verification/interpretation, 1 notation and communication.