TOPICS
Search

Hill's Differential Equation


The second-order ordinary differential equation

 (d^2y)/(dx^2)+[theta_0+2sum_(n=1)^inftytheta_ncos(2nx)]y=0,
(1)

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

If only the n=0 term is present, the equation have solution

 y=C_1sin(xsqrt(theta_0))+C_2cos(xsqrt(theta_0)).
(2)

If terms n<=1 are included, the equation becomes the Mathieu differential equation, which has solution

 y=C_1C(a,-1/2b,x)+C_2S(a,-1/2b,x).
(3)

If terms n<=2 are included, it becomes the Whittaker-Hill differential equation.


See also

Hill Determinant, Whittaker-Hill Differential Equation

Explore with Wolfram|Alpha

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.

Referenced on Wolfram|Alpha

Hill's Differential Equation

Cite this as:

Weisstein, Eric W. "Hill's Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HillsDifferentialEquation.html

Subject classifications