Exactly solvable quadratic differential equation systems through generalized inversion. (English) Zbl 1517.34020
An autonomous system of the first-order differential equations with quadratic right-hand side is considered. The system is nonlinear and its solution cannot be written in symbolic form, in general. However, there exists a subclass of quadratic systems which can be transformed to a linear system of differential equations being integrable and just such quadratic differential systems are discussed. The paper presents a detailed description of an algorithm which enables to check whether such B-transformation may be found and to construct it. As a general solution to a linear differential system with constant coefficients may be easily found the solution to the original quadratic system can be also obtained. Efficiency of the algorithm is demonstrated by three examples of two- and three-dimensional quadratic systems. It should be noted also that the algorithm may be used for analysis of the higher order quadratic systems although the calculations may become quite cumbersome.
Reviewer: Alexander Prokopenya (Warszawa)
MSC:
| 34A34 | Nonlinear ordinary differential equations and systems |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
| 34A05 | Explicit solutions, first integrals of ordinary differential equations |
Keywords:
autonomous system of quadratic differential equations; analytical solution; generalized inversion transformationReferences:
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