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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)

MSC:
 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
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