Problem 1
μ=e^{2t}; (e^{2t}y)′=e^t, so y=e^{−t}+Ce^{−2t}.
Faculty use
First-Order Linear Equations and Integrating Factors
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
μ=e^{−t}; (e^{−t}y)′=te^{−t}; y=−t−1+Ce^t.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind solving y′+p(t)y=q(t) through a product-derivative structure. Use the formulas μ=e^{∫p(t)dt}; (μy)′=μq.
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 circuits, mixing, heat transfer, finance, chemical processes, and geometric envelopes, 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.