scholarly journals A central connection problem for a normal system of linear differential equations

1984 ◽  
Vol 14 (2) ◽  
pp. 257-263 ◽  
Author(s):  
Mitsuhiko Kohno ◽  
Toshiaki Yokoyama
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
Normal System ◽  
Connection Problem ◽  
Linear Differential

scholarly journals 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

1897 ◽  
Vol 38 (1) ◽  
pp. 163-178 ◽  
Author(s):  
George Chrystal
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Complete Solution ◽  
Linear Differential Equations ◽  
Linear Functions ◽  
Normal System ◽  
Theoretical Point ◽  
Dependent Variables ◽  
Linear Differential ◽  
Ordinary Linear Differential Equations

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 formwhere fr, denotes a linear function of x1, …., xs, partly of a number of non-differential equations connecting the remainder of the variables with x1, …., xs. 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.

Sign in / Sign up

Export Citation Format

Share Document