On the Asymptotic Solution of a System of Linear Differential Equations with Small Random Coefficients

1965 ◽  
Vol 10 (3) ◽  
pp. 513-515 ◽  
Author(s):  
R. F. Mateev
Keyword(s):  
Differential Equations ◽  
Asymptotic Solution ◽  
Random Coefficients ◽  
Linear Differential Equations ◽  
Linear Differential

On Linear Perturbation of Non-Linear Differential Equations

1954 ◽  
Vol 6 ◽  
pp. 561-571 ◽  
Author(s):  
F. V. Atkinson
Keyword(s):  
Differential Equations ◽  
Asymptotic Solution ◽  
Ordinary Differential Equations ◽  
Linear Differential Equations ◽  
Linear Perturbation ◽  
Matrix System ◽  
Non Linear ◽  
Linear Differential ◽  
Vector Matrix

In the theory of the asymptotic solution or stability of ordinary differential equations most attention has been given to linear or nearly-linear cases. Investigations in this field, starting primarily with those of Kneser (7) on the equation y″ + f(x)y = 0, have by now mostly been summed up in results on the vector-matrix system dy/dx = Ay + f (y, x), where y and f denote n-vectors of functions, and A an n-by-n matrix, frequently assumed constant.

The asymptotic solution of linear differential equations of the second order for large values of a parameter

1954 ◽  
Vol 247 (930) ◽  
pp. 307-327 ◽  
Keyword(s):  
Differential Equations ◽  
Asymptotic Solution ◽  
Complex Variable ◽  
Asymptotic Expansions ◽  
Recurrence Relations ◽  
Airy Function ◽  
Second Order ◽  
Linear Differential Equations ◽  
Asymptotic Solutions ◽  
Linear Differential

Asymptotic solutions of the differential equations d2w/dz2 = {uzn +f{z)} w 0, 1) for large positive values of u, have the formal expansions P(z) 1+ Z5=1 (f> l+ ^ M y us s 5=0 IIs where P is an exponential or Airy function for n 0 or 1 respectively. The coefficients A s (z) and B s (z) are given by recurrence relations. This paper proves that solutions of the differential equations exist whose asymptotic expansions in Poincaré’s sense are given by these series, and that the expansions are uniformly valid with respect to the complex variable z. The method of proof differs from those of earlier writers and fewer restrictions are made.

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