zbMATH — the first resource for mathematics

Finite order solutions of nonhomogeneous linear differential equations. (English) Zbl 0765.34004
Consider the differential equation (1) \(f^{(n)}+A_{n-1}(z)f^{(n- 1)}+\cdots+A_ 1(z)f'+A_ 0(z)f=H(z)\), where \(A_ i(z)\), \(i=0,\dots,n- 1\) and \(H(z)\) are entire functions. In this paper the authors give the answers to the following two questions: 1. What conditions on \(A_ i(z)\), \(i=0,\dots,n-1\) and \(H(z)\) will guarantee that every solution of (1) has infinite order? 2. If (1) possesses a solution \(f\) of finite order, then how do the properties of \(A_ i(z)\) and \(H(z)\) affect the properties of \(f\)? For example, if in (1) the \(\max\{\rho(A_ 1),\dots,\rho(A_{n-1}),\;\rho(H)\}<\rho(A_ 0)<1/2\), then every solution of (1) has infinite order. Several examples are given to illustrate the results. \((\rho(A)\) denote the order of \(A(z)\)).
Reviewer: A.Klíč (Praha)

34M99 Ordinary differential equations in the complex domain
34A30 Linear ordinary differential equations and systems
34C11 Growth and boundedness of solutions to ordinary differential equations
Full Text: DOI