Problem 1
The integral term with x₀=0.
Faculty use
Forced Linear Systems and Variation of Constants
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Differentiate, substitute into x′=Ax+g, check initial data, and compare component units/physical constraints.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind combining state transition with an input history. Use the formulas x(t)=e^{A(t−t₀)}x₀+∫_{t₀}^t e^{A(t−τ)}g(τ)dτ.
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 coupled oscillators, compartment models, circuits, chemical networks, control, and mechanical 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.