Problem 1
dy/dτ=y, so the rate constant is absorbed into the time scale.
Faculty use
Nondimensionalization and Similarity
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
It reveals scale-independent behavior, reduces parameters, guides experiments, and improves numerical conditioning.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind reducing parameters and exposing dominant balances through reference scales. Use the formulas t=Tτ, y=Yz; dimensionless groups compare mechanisms.
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 experimental data, engineering design, control, reliability, environmental systems, biomedical modelling, and research communication, 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.