Phase matrix of linear differential systems. (English) Zbl 0576.34032

Došlý, Ondřej

Čas. Pěst. Mat. 110, 183-192 (1985).

Consider a linear differential equation of the second order (*) \(y''+p(x)y=0\) and let \(y_ 1,y_ 2\) be its linearly independent solutions. It is known that these solutions can be expressed in the form \(y_ 1=(| \alpha (x)|)^{-1/2}\sin \alpha (x),\) \(y_ 2=(| \alpha (x)|)^{-1/2}\cos \alpha (x),\) where \(\alpha\) (x) is the so- called phase function of (*) and k is a real constant. It is shown that solutions of the matrix differential system (**) \(Y''+P(x)Y=0\), where P(x) is a symmetric \(n\times n\) matrix, can be expressed in a similar way, where instead of trigonometric functions the so called trigonometric matrices are used. The obtained results are used to derive oscillation and nonoscillation criteria for systems (**).

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A30	Linear ordinary differential equations and systems

Keywords:

second order; phase function; trigonometric matrices