Problem 1
y=2/(1−2t), which blows up at t=1/2; the maximal interval through 0 is (−∞,1/2).
Faculty use
Threshold, Harvesting and Finite-Time Blow-Up
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
The horizontal shift f(P)−h can create or destroy equilibria, separating recovery from collapse.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind using phase lines and implicit solutions to identify tipping points and maximal intervals. Use the formulas y′=f(y)−h; y′=y² ⇒ y=y₀/(1−y₀t).
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 population, finance, radioactivity, pharmacokinetics, biological growth, and geometric families, 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.