Growth of Solutions of Complex Differential Equations with Solutions of Another Equation as Coefficients

Long, Jianren; Wu, Tingmi; Wu, Xiubi

doi:10.1007/s40315-018-0245-3

Cited by 6 publications

(2 citation statements)

References 17 publications

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“…The ideas in [12,Ch. 7.4] are profound, the methods are creative, and the results of the theory have been very useful in numerous applications to differential equations [3,5,6,7,8,16,18,19,20,21,24,27].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic integration theory for $f'' + P(z)f = 0$

Gundersen¹,

Heittokangas²,

Zemirni³

2020

Preprint

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Asymptotic integration theory gives a collection of results which provide a thorough description of the asymptotic growth and zero distribution of solutions of (*) f ′′ +P (z)f = 0, where P (z) is a polynomial. These results have been used by several authors to find interesting properties of solutions of (*). That said, many people have remarked that the proofs and discussion concerning asymptotic integration theory that are, for example, in E. Hille's 1969 book Lectures on Ordinary Differential Equations are difficult to follow. The main purpose of this paper is to make this theory more understandable and accessible by giving complete explanations of the reasoning used to prove the theory and by writing full and clear statements of the results.A considerable part of the presentation and explanation of the material is different from that in Hille's book.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic integration theory for $f'' + P(z)f = 0$

Gundersen¹,

Heittokangas²,

Zemirni³

2020

Preprint

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“…Theorem 6 Let 𝐴 𝑗 (𝑧), 𝑗 = 0,1, … , 𝑛 − 1, 𝐹(𝑧) be defined as in Theorem 5 and satisfy inequality (9). Then every solution 𝑓 ≠ 0 with 𝜌(𝑓) = ∞ of Eq.…”

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confidence: 99%

On the Growth of Solutions of Nonhomogeneous Higher order Complex Linear Differential Equations

alkhalidy

2022

IHJPAS

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The nonhomogeneous higher order linear complex differential equation (HOLCDE) with meromorphic (or entire) functions is considered in this paper. The results are obtained by putting some conditions on the coefficients to prove that the hyper order of any nonzero solution of this equation equals the order of one of its coefficients in case the coefficients are meromorphic functions. In this case, the conditions were put are that the lower order of one of the coefficients dominates the maximum of the convergence exponent of the zeros sequence of it, the lower order of both of the other coefficients and the nonhomogeneous part and that the solution has infinite order. Whiles in case the coefficients are entire functions, any nonzero solution with finite order has hyper order equals to the lower order of one of its coefficients is proved. In this case, the condition that the lower order of one of the coefficients is greater than the maximum of the lower order of the other coefficients and the lower order of the nonhomogeneous part is assumed.

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