scholarly journals The Boundary Problems and Developments Associated with a System of Ordinary Linear Differential Equations of the First Order

1923 ◽  
Vol 58 (2) ◽  
pp. 51 ◽  
Author(s):  
George D. Birkhoff ◽  
Rudolph E. Langer
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
Boundary Problems ◽  
First Order ◽  
Linear Differential ◽  
Ordinary Linear Differential Equations

Gewöhnliche lineare Differentialgleichungen n−ter Ordnung mit Distributionskoeffizienten

1980 ◽  
Vol 85 (3-4) ◽  
pp. 291-298 ◽  
Author(s):  
Rainer Pfaff
Keyword(s):  
Differential Equations ◽  
Continuous Solution ◽  
Initial Conditions ◽  
Linear Differential Equations ◽  
First Order ◽  
Integrable Functions ◽  
Leibniz Formula ◽  
Linear Differential ◽  
Ordinary Linear Differential Equations ◽  
Derivatives Of

SynopsisWe give a formula (4) for a variety of ordinary linear differential equations of order n with distributional coefficients. There appear as coefficients distributions of order k ≦ n/2, i.e. these distributions are kth distributional derivatives of locally L-integrable functions. With a suitable transformation (7) the differential equations can be transformed into first order systems (8) with integrable coefficients. From this follows the existence of a continuous solution, which can be uniquely determined by proper initial conditions.The coefficients in the differential equations considered are chosen as general as possible but such that a transformation into a system with integrable coefficients can be performed, and that all products are defined by Leibniz' formula.

scholarly journals Limit Behavior of Solutions of Ordinary Linear Differential Equations

1994 ◽  
Vol 1 (3) ◽  
pp. 315-323
Author(s):  
František Neuman
Keyword(s):  
Differential Equations ◽  
Oscillatory Behavior ◽  
Linear Differential Equations ◽  
Limit Behavior ◽  
Limit Sets ◽  
Equivalent Linear ◽  
Linear Differential ◽  
Ordinary Linear Differential Equations

Abstract A classification of classes of equivalent linear differential equations with respect to ω-limit sets of their canonical representatives is introduced. Some consequences of this classification to the oscillatory behavior of solution spaces are presented.

The Wave Theory of the Electron

1928 ◽  
Vol 24 (4) ◽  
pp. 501-505 ◽  
Author(s):  
J. M. Whittaker
Keyword(s):  
Wave Function ◽  
Differential Equations ◽  
Fourth Order ◽  
Commutative Algebra ◽  
Wave Theory ◽  
Wave Functions ◽  
Linear Differential Equations ◽  
Wave Mechanics ◽  
First Order ◽  
Linear Differential

In two recent papers Dirac has shown how the “duplexity” phenomena of the atom can be accounted for without recourse to the hypothesis of the spinning electron. The investigation is carried out by the methods of non-commutative algebra, the wave function ψ being a matrix of the fourth order. An alternative presentation of the theory, using the methods of wave mechanics, has been given by Darwin. The four-rowed matrix ψ is replaced by four wave functions ψ1, ψ2, ψ3, ψ4 satisfying four linear differential equations of the first order. These functions are related to one particular direction, and the work can only be given invariance of form at the expense of much additional complication, the four wave functions being replaced by sixteen.

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