scholarly journals Solutions of first order differential equations which are solutions of linear differential equations of higher order

1959 ◽  
Vol 10 (6) ◽  
pp. 936-936 ◽  
Author(s):  
Lawrence Goldman
Keyword(s):  
Differential Equations ◽  
Higher Order ◽  
Linear Differential Equations ◽  
First Order ◽  
Linear Differential ◽  
First Order Differential Equations

scholarly journals Application of The Riccati Method to The Adiabatic Oscillations of Stars

2004 ◽  
Vol 193 ◽  
pp. 483-486
Author(s):  
M. Takata ◽  
W. Löffler
Keyword(s):  
Differential Equations ◽  
Riccati Equation ◽  
Shooting Method ◽  
Relaxation Method ◽  
Linear Differential Equations ◽  
Matrix Riccati Equation ◽  
First Order ◽  
Nonlinear Matrix ◽  
Linear Differential ◽  
First Order Differential Equations

AbstractThe eigenmodes of the adiabatic oscillations of stars are usually calculated numerically by solving the system of the four linear first-order differential equations using either the relaxation method or the shooting method. Finding some shortcomings in these conventional methods, we adopt another method, namely the Riccati method, in which it is not the system of the linear differential equations but the nonlinear matrix Riccati equation that is solved numerically. After describing the method, we discuss its advantages and give some demonstrations.

Generalized systems of linear differential equations

1981 ◽  
Vol 89 (1-2) ◽  
pp. 1-14 ◽  
Author(s):  
Rainer Pfaff
Keyword(s):  
Differential Equations ◽  
Bounded Variation ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Differential Systems ◽  
First Order ◽  
Linear Differential Systems ◽  
Generalized Systems ◽  
Linear Differential ◽  
Derivatives Of

SynopsisWe consider ordinary linear differential systems of first order with distributional coefficients and distributional nonhomogeneous terms. Firstly the coefficients are assumed to be functions, secondly to be first order distributions (i.e. first derivatives of functions which are integrable or of bounded variation), and thirdly to be distributions of higher order.

scholarly journals Some Important Criteria for Oscillation of Non-Linear Differential Equations with Middle Term

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 346 ◽  
Author(s):  
Saad Althobati ◽  
Omar Bazighifan ◽  
Mehmet Yavuz
Keyword(s):  
Differential Equations ◽  
Comparison Method ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Middle Term ◽  
Oscillation Criteria ◽  
First Order ◽  
Non Linear ◽  
Linear Differential ◽  
Higher Order Differential Equations

In this work, we present new oscillation conditions for the oscillation of the higher-order differential equations with the middle term. We obtain some oscillation criteria by a comparison method with first-order equations. The obtained results extend and simplify known conditions in the literature. Furthermore, examining the validity of the proposed criteria is demonstrated via particular examples.

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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