Problem 1
Let y=vt: v+tv′=1+v, so v′=1/t; v=ln t+C and y=t(ln t+C).
Faculty use
Homogeneous First-Order Substitution
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
When the right side is a function of the ratio y/t (or x/y after rearrangement).
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind using v=y/t when the slope depends only on y/t. Use the formulas y′=F(y/t); y=vt, y′=v+tv′.
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 population, finance, radioactivity, pharmacokinetics, biological growth, and geometric families, 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.