On extensibility and qualitative properties of solutions to Riccati's equation (Q6639441)

The article considers the following Riccati equation \[y'=R(x)y^2+Q(x)y+P(x), \quad R(x)\equiv0\] on the real axis with continuous coefficients and non-negative discriminant of the right-hand side.\N\NThe equation under consideration has applications in many areas, in particular, in physics, financial mathematics, instrument making and mechanical engineering.\N\NThe work consists of 4 sections and a list of references. The first section is an introduction, which discusses the object of study, the history of the study of the object\N\NThe main results are presented in section 3 and their proofs are presented in section 4.\N\NAsymptotic formulas for its solutions are obtained depending on the initial values and properties of the functions representing the roots of the right-hand side of the equation. Results are obtained on the asymptotic behavior of solutions, defined near \(x\rightarrow\pm\infty\). The structure of the set of bounded solutions is studied in the case where the roots of the right-hand side of the equation are \(C_1\)-functions that are different throughout their domain of definition and monotonically tend to certain limits as \(x\rightarrow\pm\infty\).\N\NOn my opinion the results of the work are fully proven and have high scientific and practical value.