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On solutions to nonhomogeneous algebraic differential equations and their application. (English) Zbl 1307.34132

Summary: We consider solutions to the algebraic differential equation \[ f^nf'+ Q_d(z,f)= u(z) e^{v(z)}, \] where \(Q_d(z,f)\) is a differential polynomial \(f\) of degree \(d\) with rational function coefficients, \(u\) is a nonzero rational function and \(v\) is a nonconstant polynomial.
In this paper, we prove that if \(n\geq d+1\) and if it admits a meromorphic solution \(f\) with finitely many poles, then \[ f(z)= s(z) e^{v(z)/(n+ 1)}\quad\text{and}\quad Q_d(z,f)\equiv 0. \] With this in hand, we also prove that if \(f\) is a transcendental entire function, then \(f'p_k(f)+ q_m(f)\) assumes every complex number \(\alpha\), with one possible exception, infinitely many times, where \(p_k(f)\), \(q_m(f)\) are polynomials in \(f\) with degrees \(k\) and \(m\) with \(k\geq m+1\). This result generalizes a theorem originating from W. K. Hayman [Ann. Math. (2) 70, 9–42 (1959; Zbl 0088.28505)].

MSC:

34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Citations:

Zbl 0088.28505
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References:

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