一类非线性复微分方程的超越亚纯解

On Transcendental Meromorphic Solutions of a Certain Type of Nonlinear Complex Differential Equations

本文主要研究非线性复微分方程f^(4)+a(z)ff^((k))=p_(1)(z)e^(α_(1)(z))+p_(2)(z)e^(α_(2)(z))的超越亚纯解,其中a,p_(1),p_(2)是非零的有理函数,α_(1),α_(2)是非常数的多项式.进一步地,考虑当亚纯解存在时,α_(1),α_(2),p_(1)和p_(2)所满足的条件.另外,还讨论了非线性复微分方程f^(3)+a(z)f’=p_(1)(z)e^(ν(z))+p_(2)(z)e^(-ν(z))的亚纯解的存在性,其中a,p_(1),p_(2)是非零的有理函数,ν是非常数的多项式.所得的结果直接推广了一些已知的结果.

We study transcendental meromorphic solutions of a certain type of nonlinear complex differential equations f^(4)+a(z)ff^((k))-p_(1)(z)e^(α_(1)(z))+p_(2)(z)e^(α_(2)(z)),where a,p_(1),p_(2) are non-zero rational functions andα_(1),α_(2) are nonconstant polynomials.Further,we can derive the conditions concerning the termsα_(1),α_(2),p_(1) and p_(2) that are necessary for the existence and the form of a transcendental meromorphic solution of the equation above.In addition,we analyze the existence of meromorphic solutions of another type of nonlinear complex differential equations f^(3)+a(z)f’=p_(1)(z)e^(v(z))+p_(2)(z)e^(-v(z)),where a,p_(1),p_(2) are non-zero rational functions and v is a nonconstant polynomial.These results extend some known results obtained most recently.