Problem 1
Roots 2 and 3 give y=C₁e^{2t}+C₂e^{3t}.
Faculty use
Second-Order Linear Structure and Characteristic Equations
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
A second-order IVP requires two independent conditions, and the solution space is two-dimensional.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind turning constant-coefficient homogeneous equations into polynomial root problems. Use the formulas ay″+by′+cy=0; ar²+br+c=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 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.