For a given sequence of complex numbers $Λ$ the equations $f^{(n)} + Af^m = 0, m,n ∈ \mathbb{N} (f'' + Af = 0)$ were constructed, where $A$ is an entire (meromorphic in $\mathbb{C}$ ) function. This equations have meromorphic solutions with poles at the points of $Λ$. The result obtained for the equation $f'' + Af = 0$ is generalized for the case of two sequences.
[1] Šeda V. On some properties of solutions of the differential equation $y'' = Q(z)y,$ where $Q(z) \not\equiv 0$ is an entire function // Acta F.R.N. Univ. Comen. Mathem. - 1959. - 4 . - P. 223-253 (in Slovak).
[2] Bank S. A note on the zero-sequences of solutions of linear differential equations // Results in Mathematics. - 1988. - 13 . - P. 1-11.
[3] Shen L.-C. Construction of differential equation $y'' + Ay = 0$ with solutions having the prescribed zeros // Proceedings of the AMS. - 1985. - 95, N4. - P. 544-546.
[4] Heittokangas J., Laine I. Solutions of $f'' + A(z)f = 0$ with prescribed sequences of zeros // Acta Math. Univ. Comenianae. - 2005. - 74 , N2. - P. 287- 307.
[5] Bank S., Laine I. On the zeros of meromorphic solutions of second-order linear differential equations // Comment. Math. Helvetici. - 1983. - 58 , N1. - P. 656-677.
[6] Kudryashov N. A. Nonlinear differential equations of the fourth order with transcendental solutions // Theoretical and Mathematical Physics. 2000. 122, N1. P. 72-87.
[7] Kudryashov N. A. Generalizations of the Penlevé equations // Theoretical and Mathematical Physics. 2003. 137, N3. P. 408-423.
[8] Shabat B. V. Introduction to complex analysis. M.: Nauka, 1976. 320 p.
[9] Eremenko A., Liao L., Ng T. Meromorphic solutions of higher order Briot-Bouquet differential equations // Math. Proc. Camb. Phil. Soc. - 2009. - 146 . - P. 197-206.
- ACS Style
- Lukivska, .V.; Shavala, O. On meromorphic solutions of differential equations with given poles. Bukovinian Mathematical Journal. 2016, 3
- AMA Style
- Lukivska V, Shavala O. On meromorphic solutions of differential equations with given poles. Bukovinian Mathematical Journal. 2016; 3(2).
- Chicago/Turabian Style
- Dzvenislava Volodymyrivna Lukivska, Olena Shavala. 2016. "On meromorphic solutions of differential equations with given poles". Bukovinian Mathematical Journal. 3 no. 2.