Problem 1
Let x₁=y,x₂=y′. Then x₁′=x₂ and x₂′=−2x₁−3x₂.
Faculty use
From Higher-Order Equations to First-Order Systems
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
It supports matrix methods, numerical solvers, phase portraits, and multi-component models.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind introducing state variables to rewrite one nth-order equation as n first-order equations. Use the formulas x₁=y, x₂=y′; x′=[x₂, F(t,x₁,x₂)].
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.