Problem 1
u′=1+y′=u²; hence u=−1/(t+C), so y=−t−1/(t+C).
Faculty use
Substitutions for Shifted and Composite Slopes
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Back-substitute, state excluded values, verify by differentiation, and apply initial data on a valid interval.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind choosing a new variable that collapses repeated combinations such as at+by. Use the formulas u=at+by; u′=a+by′.
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.