Stability and Instability, Exact and Approximate Solutions of Linear Differential Equations with Time Varying Coefficients

摘要

The familiar soluble linear differential equations and the properties of linear systems are reviewed. The importance of linear differential equations and the origins of some such problems are discussed. The exact solutions of a class of second order differential equations and a more general class of nth order equations - those for which the fundamental solution matrix is composed of orthogonal vectors - are derived. A canonical form and the transformation, which leads to it, are derived and from this canonical form conclusions are drawn with respect to stability or instability using a modification of Liapounov stability theory. The canonical form dx/dt = (lambda + N)x where lambda is a diagonal matrix and N is an nxn skew symmetric matrix, is subsequently used to obtain approximate solutions to both homogeneous and nonhomogeneous linear differential equations. While the methods of this dissertation may theoretically be applied to an nth order linear system, in practice they are limited to second and third order systems unless machines are used. (Author)