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Periodic solutions of the Riccati differential equation with periodic coefficients. (English) Zbl 1269.34094

The paper investigates the existence, the number and the growth of periodic meromorphic solutions of the Riccati differential equation, \[ w'= a(z)+ b(z)w+ c(z)w^2 \] with \(a(z)\), \(b(z)\) and \(c(z)\) meromorphic functions. Using the transformation \[ w= {1\over c(z)} u-{b(z)\over 2c(z)}- {c'(z)\over 2c(z)^2}, \] where \(c(z)\neq 0\), and this equation can be transformed to the normal form, \[ u'= A(z)+ u^2, \] where \[ A= ac-{b^2\over 4}+ {b'\over 2}- {3\over 4}\Biggl({c'\over c}\Biggr)^2- {bc'\over 2c}+ {c''\over 2c}. \] Here, \(A(z)\) is a periodic function with period \(\omega\). Several excellent and well-written proofs are given for the existence and the number and growth of the periodic meromorphic solutions. Several examples are given, one reads as follows: \[ u'= -{1\over 4} e^{2z} e^{2e^z}- {1\over 4} e^{2z}-{1\over 4}+ u^2 \] has several meromorphic solutions.

MSC:

34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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