Hill's Differential Equation

The second-order ordinary differential equation

(1)

where  are fixed constants. A general solution can be given by taking the "determinant" of an infinite matrix.

If only the  term is present, the equation have solution

(2)

If terms  are included, the equation becomes the Mathieu differential equation, which has solution

(3)

If terms  are included, it becomes the Whittaker-Hill differential equation.

REFERENCES:

Hill, G. W. "On the Part of the Motion of Lunar Perigee Which is a Function of the Mean Motions of the Sun and Moon." Acta Math. 8, 1-36, 1886.

Ince, E. L. Ordinary Differential Equations. New York: Dover, p. 384, 1956.

Magnus, W. and Winkler, S. Hill's Equation. New York: Dover, 1979.

Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 123, 1997.