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The general form of the Sturm-Liouville system is

where .

Bessel Functions: For , , , , , and , the Sturm-Liouville equation becomes the Bessel's differential equation

which is defined on . The solutions of the Bessel's differential equation is called Bessel Functions of the first kind which form a complete orthogonal set on the interval with respect to . (Further detail; see plots.)

Legendre Polynomials: For , , , , , and , the Strangleholds equation becomes the Legendre's differential equation

which is defined on . The solutions of the Legendre's differential equation with is called Legendre Polynomials which form a complete orthogonal set on the interval . (Further detail; see plots.)

Hermite Polynomials: For , , , , , and , the Sturm-Liouville equation becomes the Hermite's differential equation

which is defined on . The solutions of the Hermite's differential equation with is called Hermite Polynomials which form a complete orthogonal set on the interval with respect to . (Further detail; see plots.)

Laguerre Polynomials: For , , , , , and , the Sturm-Liouville equation becomes the Laguerre's differential equation

which is defined on . The solutions of the Laguerre's differential equation with is called Laguerre Polynomials which form a complete orthogonal set on the interval with respect to . (Further detail; see plots.)

Chebyshev Polynomials: For , , , , , and , the Sturm-Liouville equation becomes the Chebyshev's differential equation

which is defined on . The solutions of the Chebyshev's differential equation with is called Chebyshev Polynomials which form a complete orthogonal set on the interval with respect to . (Further detail; see plots.)