Problem 1
The family slope is y′=2y/x; orthogonal slope is −x/(2y), so y dy=−x dx/2 and x²+2y²=C.
Faculty use
Orthogonal Trajectories and Family Geometry
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
The slope field must be written in x and y so every member of the family is represented by one differential equation.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind finding curves that intersect a given family at right angles. Use the formulas m_⊥=−1/m; replace y′ by −1/y′ then solve.
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.