THE BOUNDARY PROBLEMS AND DEVELOPMENTS 



ASSOCIATED WITH A SYSTEM OF ORDINARY 



LINEAR DIFFERENTIAL EQUATIONS OF 



THE FIRST ORDER tmiTntiKA - 



■ i ff yMT 

 By George D. Birkhoff and Rudolph E. Langer.i 



Introduction. 



It is the purpose of this paper to develop in outline the theory of a 

 system of n ordinary linear differential equations of the first order 

 containing a parameter and subject to certain boundary conditions. 

 Toward this end the notation of matrices is used. For the convenience 

 of the reader the paper opens with a brief review of the fundamentals 

 of matrix algebra and the integration and differentiation of matrices. 

 This is followed by an expository discussion of the homogeneous and 

 non-homogeneous differential matrix equations of the first order. 

 The major portion of the treatment is devoted, however, to the homo- 

 geneous differential vector equation with a complex parameter in its 

 coefficient, and to the system composed of such an equation and 

 suitable boundary conditions. The solutions of the equation for large 

 values of the parameter are discussed and the formal development of 

 a vector of arbitrary functions into a series of solutions of the system 

 is obtained. The paper closes with the proof of the convergence of 

 this development under appropriate conditions, which, in the ordinary 

 notation, establishes the possibility of simultaneously expanding n 

 arbitrary functions in terms of the characteristic solutions of a prop- 

 erly restricted differential system of the type 



n 



y'i(x) = 2 [a ik (x)\ + b ik (x)}y k (x), 



n 



2 {a ik y k (a) + &#*(&)} =0, i = 1, 2,. . .n 2 



When reduced to a single equation of the nth. order this includes as a 

 special case the expansions obtained by Birkhoff in 1908. 



1 Much of the material preceding the proof of convergence is due to Birk- 

 hoff, having been developed by him in lectures at Harvard University in the 

 fall of 1920. The reorganization of this material into its present form, the 

 treatment of the irregular case, and the proof of convergence are due to Langer. 



2 For other developments in this field and more complete references see the 

 following papers in the Transactions of the American Mathematical Society: 



Birkhoff, On the Asymptotic Character of the Solutions of Certain Linear 



