On the solving of the Hill's differential equation through the method of operational calculus

Abstract

Using a method previously applied to the treatment of the Mathieu differential equation, we solve the Hill's differential equation of lunar theory through the way of operational calculus, which avoids the cumbersome infinite determinants of the classical procedure. The one-sided Laplace transformation changes it into a difference equation with an infinite number of terms and variable coefficients. When its first member is divided by a suitable factor, this difference equation is the image of an integral equation of the Volterra type which is equivalent to the initial Hill's differential equation. Solution of this Volterra integral equation is unique and it is the general solution of the Hill's differential equation. This solution is a series in the powers of a small dimensionless parameter∈2 which appears as a factor in the second member of the Hill's differential equation. We reduce it to the sum of its terms of degree ≤12 with respect to ɛ which is the precision usually required in a lunar theory and we write down effectively the coefficients of the terms in∈2, (∈2)2 and the coefficient of the term in (∈2)3 which depends upon the initial valuey(0) of the Hill's differential equation.