Existence of transcendental meromorphic solutions on some types of nonlinear differential equations
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We show that when $n>m$, the following delay differential equation \begin{equation*} f^n(z)f'(z)+p(z)(f(z+c)-f(z))^m=r(z)e^{q(z)} \end{equation*} of rational coefficients $p,r$ doesn't admit any transcendental entire solutions $f(z)$ of finite order. Furthermore, we study the conditions of $\alpha_1, \alpha_2$ that ensure existence of transcendental meromorphic solutions of the equation \begin{equation*} f^n(z) + f^{n-2}(z)f'(z)+ P_d(z,f)=p_1(z)e^{\alpha_1( z)}+p_2(z)e^{\alpha_2 (z)}. \end{equation*} These results have improved some known theorems obtained most recently by other authors.
Abstract
We show that when $n>m$, the following delay differential equation \begin{equation*} f^n(z)f'(z)+p(z)(f(z+c)-f(z))^m=r(z)e^{q(z)} \end{equation*} of rational coefficients $p,r$ doesn't admit any transcendental entire solutions $f(z)$ of finite order. Furthermore, we study the conditions of $\alpha_1, \alpha_2$ that ensure existence of transcendental meromorphic solutions of the equation \begin{equation*} f^n(z) + f^{n-2}(z)f'(z)+ P_d(z,f)=p_1(z)e^{\alpha_1( z)}+p_2(z)e^{\alpha_2 (z)}. \end{equation*} These results have improved some known theorems obtained most recently by other authors.
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