On some properties of first order algebraic differential equations (Q2323138)

The main result of the paper is the following statement Theorem. Let \(B(z,{\omega_1},{\omega_2})\) be an irreducible polynomial in the ring \(\mathbb{C}[z,{\omega_1},{\omega_2}]\), \[ \begin{aligned} & B=\sum_{m=0}^{N}{\sum_{n=0}^{d}{{b_{mn}}(z)\omega_1^{m}\omega_2^{n}}}\text{ and }\frac{\partial B}{\partial {\omega_2}}\ne 0; \\ & \Delta_{B}=\frac{\partial}{\partial z}+{\omega_2}\frac{\partial}{\partial {\omega_1}}-\frac{\frac{\partial B}{\partial z}+\frac{\partial B}{\partial {\omega_1}}{\omega_2}}{\frac{\partial B}{\partial {\omega_2}}} \frac{\partial}{\partial {\omega_2}}. \end{aligned} \] Let \(f(z)\) be an entire function of finite order that is not a polynomial and such that \(B(z,f(z),{f}'(z))\equiv 0\) in \(\mathbb{C}\). Then for any set of different natural numbers \({i_1},\dots,{i_{d+1}}\), there are polynomials \(\{a_{i_k}(z),k=1,\dots,d+1\}\) and the polynomial \(a_{i_0}(z)\) with the following properties: \begin{itemize} \item[1)] the coefficients of each of \(\{a_{i_1}(z),\ldots ,a_{i_{d+1}}(z)\}\) are rational functions with complex coefficients of the coefficients of the polynomials \(\{b_{mn}(z)\}\); \item[2)] \(\sum_{k=1}^{d+1}a_{i_k}(z)\Delta_{B}^{i_k}(\omega_1)-a_{i_0}(z)\omega_1\in \bmod (B);\) \item[3)] the function \(y=f(z)\) satisfies the linear differential equation \[ \sum_{k=1}^{d+1}a_{i_k}(z)y^{(i_k)}-a_{i_0}(z)y=0\ (1); \] \item[4)] if the point \(\left( {z_0},\omega_1^0,\omega_2^0 \right)\in \mathbb{C}^{3}\) is such that \[ B\left( {z_0},\omega_1^0,\omega_2^0 \right)=0,\quad \frac{\partial B}{\partial {\omega_2}}\left( {z_0},\omega_1^0,\omega_2^0 \right)\ne 0 \] and the function \(y=\varphi(z)\) satisfies the equation \(B(z,\varphi (z),{\varphi}'(z))\equiv 0\) in some nonempty circle \(K=\{|z-{z_0}|<\delta \},\) moreover, \(\varphi ({z_0})=\omega_1^0,\varphi'({z_0})=\omega_2^0\), then \(y=\varphi (z)\) also satisfies equation (1) in the circle \({K_1}=\{|z-{z_0}|<{\delta_1}\}\) (for some \({\delta_1}>0\)). \end{itemize} To become acquainted with the class of problems to which this result applies, the authors recommend a monograph [\textit{V. N. Gorbuzov}, Integral solutions of algebraic differential equations. Grodno: GrSU (2006)].