Synopses & Reviews
Synopsis
Excerpt from Hill's Equation, Vol. 1: General Theory
Any homogeneous linear differential equation of second order with real periodic coefficients can be reduced to an equation of Hill's type. The specific question which arises in the theory of Hill's equation is the problem of the existence of periodic solutions. This problem has many features in common with the ordinary sturm-liouville problems, and in certain cases it canjin fact, be reduced to ordinary boundary value problems of the sturm-liouville type, (see Section However, in general such a reduction is not possible, and imposing the periodicity requirements on a solution of the differential equation leads to phenomena different from those resulting from the imposition of a homogeneous boundary condition of the sturm-liouville type. Thus, the differential equation can have two linearly independent periodic solutions but it cannot have two linearly independent solutions satisfying the same homogeneous boundary conditions. Futhermore, the value of the period of the solution (which is a multiple of the period p of the coefficients) plays an important role in the discussion of periodic solutions. In a certain sense, only the solutions of period p and 2 p are of interest. 'we shall now proceed with a detailed presentation of some basic theorems and their proofs.
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