This thesis explores the question: “Will the solution curves of a linear second-order ordinary differential equation continue crossing the x-axis, or will they eventually remain on one side?”
The analysis is focused on the Sturm–Liouville form:
-(p(x) y′)′ + q(x) y = 0
and builds upon the classical developments from Sturm’s foundational 19th-century work through to advanced criteria formulated in the 20th century.
- Oscillatory vs Nonoscillatory (Disconjugacy) – Describes whether a solution has infinitely many zeros or not.
- Sturm’s Separation Theorem – Details the interlacing behavior of zeros between two independent solutions.
- Sturm’s Comparison Theorem – Compares the zero counts of two equations by evaluating their coefficients.
- Sturm-Picone Comparison Theorem – Generalizes Sturm’s theorem by accounting for different leading coefficients.
- Leighton’s Comparison Theorem – Offers a broader zero-comparison framework using an integral identity.
- Leighton’s Criterion – Establishes an integral condition that ensures oscillation occurs.
- Kreith’s Criteria – Uses quadratic-integral conditions to test for disconjugacy.
- Levin’s Comparison Theorem – Applies a Riccati transformation to estimate zeros without needing a direct comparison equation.
- Wintner’s Criteria – Provides conditions based on integral growth at infinity to determine oscillatory behavior.
- Hartman’s Criterion – Identifies the exact threshold between oscillatory and nonoscillatory behavior.
- Structural vibration – Helps predict node positions and natural frequencies in beams, strings, and membranes without solving the complete eigenvalue problem.
- Electrical and mechanical oscillators – Evaluates whether systems will exhibit sustained oscillation or quickly settle.
- Quantum mechanics – Connects the count of bound states in a potential well to the zeros of associated wavefunctions.
- Rapid qualitative analysis – Equips science and engineer students with fast, coefficient checks for system stability and resonance.