| Ordinary differential equations are developed along with calculus,which grows in the development process of production practice and mathematics and contains a rich mathe-matical thinking method.It shows great functionality in astrophysics and other mechanics.Newton confirms that the Earth's orbit around the sun is an ellipse by solving the differen-tial equation.Neptune's existence is astronomers first calculated by the differential equation method and then actually observed.The formation and development of the ordinary differ-ential equation is also closely related to the development of mechanics,astronomy,physics and other science and technology.The development of other branches of mathematics,such as complex function,Lie group,combination topology and so on,have developed a pro-found impact on the development of ordinary differential equations.At present,the ordinary differential equation in all natural sciences and many social sciences have a wide range of applications.Can predict with the development of social technology and demand ordinary differential equation will have greater development.The development of ordinary differential equations has gone through four stages:The first stage is to find the solution as the main content of the classic theoretical stage.In 1690,Bernoulli James studied the "isochronal curve problem associated with the pendulum move-ment(in the same time,making the pendulum a complete vibration along the curve(regard-less of the size of the arc length that the pendulum experienced))".He established the model of the ordinary differential equation by analyzing and solved the equation of the cycloid with the separation variable method.In 1690,Bernoulli James proposed the "catenary prob-lem(the curve in which the rope was suspended at two fixed points(the rope was soft but not elongated))".Bernoulli John and Leibnitz solved the catenary problem with the calculus method.And later studied the isosceles trajectory problem,orthogonal trajectory problems and so on.In 1691,Leibnitz gave the variable separation method.In 1694,he used the con-stant method to convert the first-order ordinary differential equation into integral,and found that the envelope of a solution is also a solution.In 1695,Bernoulli John gave the famous Bernoulli equation,Leibnitz transforms it into a linear equation.1715-1718,Taylor discuss-es the singular solution of the differential equation,the envelope and the variable substitution formula.In 1734,Clairaut studied the Clairaut equation and found that the general solution of the equation is a linear family,and the straight line envelope is odd solution.Clairaut and Euler have carried on the comprehensive research to the singular solution,given the singular solution from the differential equation itself method.In 1734,Euler gave the definition of the proper equation.He and Claire found that the equation was the condition of the prop-er equation and found that if the equation was appropriate,the equation was integrable.In 1739,Clayrow put forward the concept of integral factor,Euler determined the equation can be used to solve the equation.In 1772,Laplace extended the concept of singularity to the equations of the higher order equation and the three variables.In 1774,Lagrange made a systematic study of the relationship between the singular solution and the general solution.He gave the general method and the singular solution to the geometric interpretation of the family of the integral curve family.The second stage is the theoretical stage of the qualitative theory of the solution.This period is a period of change in the history of mathematics,the basis of mathematical analysis,the concept of group,the creation of complex function and so on in this period,ordinary differential equation by these new concepts and new method-s,into its development the second stage.The main results of this phase are:In the 1920s,Cauchy established the existence and uniqueness theorem of the Cauchy problem.In the 1920s,Cauchy established the existence and uniqueness theorem of the Cauchy problem.In 1873,Lipschitz proposed the famous "Lipschitz condition",and improved the existence the only theorem of Cauchy.1875 and 1876 Cauchy,Lipschitz,Piana and Bika has given the ordinary differential equation successive approximation method and so on.The third stage is the analytic theory stage of the development of ordinary differential equations.One of the main results of this phase is the use of power series and generalized power series solution,find some important second order linear equations of the series solution,and get extremely important some special function,Riemann-Fuchs singularity theory is also a very important achievement at this stage.The fourth stage is the qualitative theoretical stage of the ordinary differential equation,Poincare and Lyapunov created the differential equation qualitative the-ory and differential equation motion stability theory respectively.Polynomial differential system is a simple and important ordinary differential equation,the study of the limit cycle problem occupies a very important position in the qualitative theory of differential equations,in 1900,the second half of the sixteenth question raised by Hilbert was the discussion of the maximum number and relative position of the limit cycle of the planar polynomial system.Homogeneous polynomial differential system as an important class in polynomial sys-terns.So far,homogeneous polynomial differential system has a lot of results,Markus s-tudied the classification of(P,Q)for the quadratic homogeneous polynomial vector field of P and Q.Algaba obtains the standard type of homogeneous polynomial differential system and the system invariant invariant theory.Cima obtains the classification theorem and the algebraic classification of the fourth order binary on real domain.Quasi-homogeneous polynomial differential system is a generalization of homogeneous polynomial differential system.In recent years,the quasi-homogeneous system is concerned by many scholars,for example:quasi-homogeneous decomposition,quasi-homogeneous polynomial system integrable,central problem,limit cycles,standard type and so have achieved rich results.In 2013,Garcia gives an algorithm for quasi-homogeneous classifi-cation of planar quasi-homogeneous polynomial systems,and uses this algorithm to obtain all quasi-homogeneous distributions of planar 2 and 3 polynomial systems classification.In this paper,we consider the quasi homogeneous decomposition of the Newton motion equations f(q1,q2)and g(q1,q2)are n and m order polynomials of q1,q2,respectively.We first discuss some quasi-homogeneous properties of the system(1),and then give the quasi-homogeneous classification algorithm of(1).Finally,we use the algorithm to give when m ? n = 4 the classification of quasi homogeneous of(1),write ?=(s1,s2,s3,s4,d),quasi homogeneous vector field(pi,p2,f,g).The list of classification results is as follows:... |
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