Jacobi Differential Equation

(1)

or

(2)

The solutions are Jacobi polynomials  or, in terms of hypergeometric functions, as

(3)

The equation (2) can be transformed to

(4)

where

(5)

and

(6)

where

(7)

Zwillinger (1997, p. 123) gives a related differential equation he terms Jacobi's equation

(8)

(Iyanaga and Kawada 1980, p. 1480), which has solution

(9)

Zwillinger (1997, p. 120; duplicated twice) also gives another types of ordinary differential equation called a Jacobi equation,

(10)

(Ince 1956, p. 22).

In the calculus of variations, the partial differential equation

(11)

where

(12)

is called the Jacobi differential equation.

REFERENCES:

Bliss, G. A. Calculus of Variations. Chicago, IL: Open Court, pp. 162-163, 1925.

Ince, E. L. Ordinary Differential Equations. New York: Dover, p. 22, 1956.

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1480, 1980.

Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997.