Problem 1
m(m−1)−m+1=(m−1)²=0, so y=x(C₁+C₂ln x).
Faculty use
Euler–Cauchy Equations
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Let x=e^t (t=ln x).
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind using power-law trial solutions or logarithmic change of variable. Use the formulas ax²y″+bxy′+cy=0; trial y=x^m.
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.