Special Differential Equations

A handful of named ODEs recur throughout physics and engineering. Learn Legendre's equation and polynomials, Bessel's equation and functions, Hermite, Laguerre and Chebyshev equations, where each arises (spheres, cylinders, quantum oscillators, approximation), and key properties like orthogonality.

Separating the heat or wave equation in cylindrical coordinates gives Bessel's equation; in spherical coordinates, Legendre's. Their solutions describe drum vibrations, waveguide modes, fins, antenna patterns and electron orbitals.

An analogy

Just as sine and cosine are the natural shapes for vibrations on a straight string, Bessel functions are the natural shapes for vibrations on a round drum.

Why names?

Each equation's solutions have been studied, tabulated and implemented (scipy.special, MATLAB besselj). Recognising the equation saves solving it from scratch.

Bessel functions

J₀ behaves like a decaying cosine; its zeros (2.405, 5.520, 8.654…) give drum-head and circular-waveguide mode frequencies. Y_ν is singular at x = 0 and is dropped for solid regions.

Legendre polynomials

Arise from Laplace's equation in spherical coordinates; describe potentials around spheres and multipole expansions. Orthogonal on [−1, 1].

Other families

Hermite: quantum harmonic oscillator and Gaussian beams. Laguerre: hydrogen atom radial functions. Chebyshev: minimax polynomial approximation and filter design (Chebyshev filters).

Formulas

• Legendre: (1 − x²)y″ − 2xy′ + n(n + 1)y = 0; P₀ = 1, P₁ = x, P₂ = (3x² − 1)/2
• Bessel: x²y″ + xy′ + (x² − ν²)y = 0; y = c₁J_ν(x) + c₂Y_ν(x)
• Hermite: y″ − 2xy′ + 2ny = 0
• Chebyshev: (1 − x²)y″ − xy′ + n²y = 0; Tₙ(cos θ) = cos nθ

Worked example: Drum fundamental

Given: Circular membrane radius a, wave speed c

Find: Fundamental frequency.

• Mode shape J₀(kr) with J₀(ka) = 0 → ka = 2.405
• f = 2.405c/(2πa)

Answer: f₀₁ = 0.383 c/a

Worked example: Legendre check

Given: P₂ = (3x² − 1)/2, n = 2

Find: Verify it satisfies Legendre's equation.

• y′ = 3x, y″ = 3
• (1 − x²)3 − 2x(3x) + 6(3x² − 1)/2 = 3 − 3x² − 6x² + 9x² − 3 = 0 ✓

Answer: Satisfied

Common mistakes

• Wrong: Keeping Y_ν in a solid disc. — Right: Y_ν → −∞ at r = 0; set its coefficient to 0.
• Wrong: Using Legendre outside [−1, 1] for angles. — Right: x = cos θ lies in [−1, 1].

Applications

• Waveguides & fibres: Bessel modes in circular guides.
• Electrostatics: Legendre expansions around spheres.
• Filters & approximation: Chebyshev polynomials.

Key takeaways

• Recognise the named equation.
• Bessel ↔ cylinders; Legendre ↔ spheres.
• Zeros give mode frequencies.
• Orthogonality enables series expansions.

Going further

• Sturm–Liouville theory: Unifies orthogonality of these families (M24).
• Modified Bessel functions: I_ν, K_ν for fins and diffusion.
• Spherical harmonics: Y_l^m from Legendre functions.