Problem 1
Repeated root r=2 gives y=(C₁+C₂t)e^{2t}.
Faculty use
Repeated Roots and Generalized Modes
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
A repeated root supplies only one exponential eigenmode; multiplication by t creates an independent solution.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind constructing a second solution when the characteristic root repeats. Use the formulas (r−r₀)² ⇒ y=(C₁+C₂t)e^{r₀t}.
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 mechanical vibration, structural response, circuits, feedback, and transient engineering systems, 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.