On Meromorphic Solutions of Non-linear Differential Equations and Their Applications

Abstract

In this paper, the authors consider meromorphic solutions of nonhomogeneous differential equation

f^n(f^{\mathrm{\prime }}+af)+P_d(z,f)=u(z){\mathrm{e}}^{v(z)},

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f^{n}(f^{\prime}+af)+P_{d}(z,f)=u(z){\rm{e}}^{v(z)},$$\end{document}

where n is a positive integer, a is a nonzero constant, P_d(z, f) is a differential polynomial in f(z) of degree d with rational functions as its coefficients and d ≤ n − 1, u(z) is a nonzero rational function, v(z) is a nonconstant polynomial with v′(z) ≠ (n + 1)a, v′(z) ≠ −na and

v^{\mathrm{\prime }}(z)\ne -\frac{(n+1{)}^2}{n}a

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$v^{\prime}(z)\ne-{(n+1)^{2}\over{n}}a$$\end{document}

. They prove that if it admits a meromorphic solution f(z) with finitely many poles, then

\begin{array}{ccc}f(z)=s(z){\mathrm{e}}^{\frac{v(z)}{n+1}}& \text{and}& P_d(z,f)\equiv 0,\end{array}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\matrix{{f(z) = s(z){{\rm{e}}^{{{v(z)} \over {n + 1}}}}} & {{\text{and}}} & {{P_d}(z,f) \equiv 0,}}$$\end{document}

where s(z) is a rational function and sⁿ[(n + 1)s′ + sv′] + (n + 1)asⁿ⁺¹ = (n + 1)u. Using this result, they also prove that if f(z) is a transcendental entire function, then fⁿ(f′ + af) + q_m(f) assumes every complex number α infinitely many times, except for a possible value q_m(0), where n, m are positive integers with n ≥ m + 1 and q_m(f) is a polynomial in f(z) with degree m.

Keywords

Non-linear differential equations / Differential polynomial / Meromorphic functions / Entire functions / Nevanlinna theory / Normal family / 30D35

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