Problem 1
x²y″+xy′+(x²−ν²)y=0.
Faculty use
Bessel and Legendre Foundations
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
For nonnegative integer n in Legendre’s equation, with an appropriate normalization.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind recognizing canonical special-function equations from radial and angular models. Use the formulas x²y″+xy′+(x²−ν²)y=0; (1−x²)y″−2xy′+n(n+1)y=0.
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 vibration modes, radial heat flow, quantum models, waveguides, orbital problems, and approximation theory, 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.