Expressions of meromorphic solutions of a certain type of nonlinear complex differential equations
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In this paper, the expressions of meromorphic solutions of the following nonlinear complex differential equation of the form $$f^{n}+Q_{d}(z,f)=\sum_{i=1}^{3}p_{i}(z)e^{\alpha_{i}(z)}$$ are studied by using Nevanlinna theory, where $n\geq5$ is an integer, $Q_{d}(z,f)$ is a differential polynomial in $f$ of degree $d\leq n-4$~with rational functions as its coefficients, $p_{1}(z)$, $p_{2}(z)$, $p_{3}(z)$~are non-vanishing rational functions, and $\alpha_{1}(z)$, $\alpha_{2}(z)$, $\alpha_{3}(z)$ are nonconstant polynomials such that $\alpha_{1}'(z)$, $\alpha_{2}'(z)$, $\alpha_{3}'(z)$ are distinct each other. Moreover, examples are given to illustrate the accuracy of the condition.
Abstract
In this paper, the expressions of meromorphic solutions of the following nonlinear complex differential equation of the form $$f^{n}+Q_{d}(z,f)=\sum_{i=1}^{3}p_{i}(z)e^{\alpha_{i}(z)}$$ are studied by using Nevanlinna theory, where $n\geq5$ is an integer, $Q_{d}(z,f)$ is a differential polynomial in $f$ of degree $d\leq n-4$~with rational functions as its coefficients, $p_{1}(z)$, $p_{2}(z)$, $p_{3}(z)$~are non-vanishing rational functions, and $\alpha_{1}(z)$, $\alpha_{2}(z)$, $\alpha_{3}(z)$ are nonconstant polynomials such that $\alpha_{1}'(z)$, $\alpha_{2}'(z)$, $\alpha_{3}'(z)$ are distinct each other. Moreover, examples are given to illustrate the accuracy of the condition.
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