Problem 1
It defines the weighted inner product and eigenfunction orthogonality.
Faculty use
Sturm–Liouville Structure and Orthogonality
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
They convert a field into independent modal coordinates satisfying simpler time equations.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind placing second-order equations in self-adjoint form and using weighted inner products. Use the formulas −(py′)′+qy=λwy; ∫w y_m y_n=0 for m≠n.
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 heat conduction, vibrating strings, structural modes, diffusion, electrostatics, acoustics, and field simulation, 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.