Problem 1
Substitute into the ODE, measure residual on the interval, compare successive truncations and a numerical solution.
Faculty use
Series Truncation, Numerical Comparison and Verification
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Coefficients and convergence depend on the chosen center and nearby singularities.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind evaluating partial sums, residuals, and convergence evidence. Use the formulas residual R_N=L[y_N]−g; compare N, N+2 and numerical solver.
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.