II.—A Fundamental Theorem regarding the Equivalence of Systems of Ordinary Linear Differential Equations, and its Application to the Determination of the Order and the Systematic Solution of a Determinate System of such Equations

Extract

Systems of ordinary linear differential equations are of great importance, both from a practical and from a theoretical point of view. They figure largely in dynamical problems; and Jacobi has shown that the general problem of determining the order of any system of ordinary differential equations whatever can be reduced to the problem of determining the order of a linear system with constant coefficients. Nevertheless, the present state of the theory of such a system still leaves something to be desired. It is true that a logical and systematic process for the solution was given by Cauchy. This consists in first replacing the system by another in which only first differential coefficients occur, by introducing as auxiliary variables the successive differential coefficients of the various dependent variables up to the highest but one, and then reducing this system to the “normal form” by calculating the differential coefficients as linear functions of the dependent variables. It happens, however, when we attempt to do this, that we are led to a system consisting partly of differential equations of the form

where f_r, denotes a linear function of x₁, …., x_s, partly of a number of non-differential equations connecting the remainder of the variables with x₁, …., x_s. The order of the system—that is to say, the number of independent arbitrary constants required for its complete solution—is the number of differential equations in the normal form; but no rule is readily deducible from the method for determining beforehand how many of the equations in the normal system will be differential equations, so that we cannot predict the order of the system without actually going through the labour of reduction. Moreover, the normal form is in practice often not the most convenient for the purposes of solution.