微分方程-1

     我把方程分为两大类：函数方程（这个“数”不止是实数，还可以是复数、矩阵、甚至张量、四元数等等）、逻辑方程（即非传统的数类方程）。而函数方程有可细分为代数方程、超越方程、矩阵方程、微分（积分）方程、泛函微分方程、含差分的微分方程、通常的函数方程（包括迭代在内）等。我们都知道代数方程中五次或者以上的没有一般形式的公式解，超越方程基本只能数值求解，矩阵方程的情况和“数”的方程差不多，而通常的函数方程除了一些技巧以外，大部分只能用级数法求解。最后的微分（积分）方程也不是很乐观，并不是都有可积的解（而且绝大多数都是不可解的）。

     对于微分方程中的常微分方程，本课主要研究的是一些常见可积类型的求解法、解的定性法、数值求解、级数求解、数学变换求解、微分方程在几何以及物理问题中的应用等。

一，基本概念与初等积分法。

1， 微分方程的基本概念、相空间、积分曲线、具有一维相空间的微分方程。

2， 具有多维相空间的微分方程、相曲线、后继函数、Poincare映射、小振动、解的存在性与唯一性、Lipscitz条件。

3， 可分离变量的方程、两个系统的笛卡尔积。Lotka-Volterra模型、平衡位置、一阶线性齐次方程、它的对称群。具有周期系数的一阶线性齐次方程。 

3，一阶线性方程，单值变换与周期系数线性方程的周期解。

4， 一阶线性非齐次方程、叠加原理、Green函数、具有周期系数的一阶线性非齐次方程、单参数微分同胚群、向量场、相流。

5， 极限环、相流上的微分同胚作用、齐次方程、拟齐次方程。

4，完全可微方程，一个自由度的Hamilton方程，摆。

6， 初等积分法、奇解。Bernoulli方程、Riccati方程、恰当型方程、位势函数、积分因子、相平面、相轨。

二，存在性定理。

7， 广义Lotka-Volterra模型、正则线元、奇解、包络、Clairaut方程、D’Alembert方程、Banach空间、逐次逼近法、压缩映射原理。

8， 向量积分、可微性与Lipscitz条件、存在性与唯一性定理的证明、Peano存在定理、等度连续、Ascoli-Arzela定理、Euler折线法。

7，初值问题的解的存在性、唯一性与连续独立性定理，Picard方法。

8，解的Picard逼近法。

9， 逐次逼近的发散、适定性问题、初值问题解的连续与可微依赖性定理、参数的连续与可微依赖性定理、延拓定理、向量场的直化。

10， 高阶微分方程与一阶微分方程组的关系、高阶微分方程的存在性与唯一性、高阶微分方程的可微性与延拓定理、微分方程组的相空间的维数、接触结构、变分方程、自治系统。

9，积分曲线与相曲线的连续性定理及其在线性方程组中的应用。

11， 闭相曲线、线性算子的单参数群、常系数线性方程的基本定理、算子的

行列式、算子的迹、Liouville公式、Liouville-Ostrogradskii公式，可对角化算子、特征方程、有摩擦力的摆方程的相曲线。

三，任意阶常系数线性方程。

12， 具有复相空间的线性微分方程、奇点的分类、特征方程具有单根的线性方程的通解。

11，齐次方程与特殊右半平面方程。

13， 用Jordan标准型求解常系数线性微分方程、线性微分方程的解空间、非齐次线性微分方程的解、复数振幅法、共振。

微分方程-2

一，线性方程组。

1，相流，线性算子的展开。

2，复化与实化，值数的计算。

3，复数展开。

4，Jordan块的展开。

1， 变系数齐次线性微分方程、变系数齐次线性微分方程的解的先验估计、变系数齐次线性微分方程的解空间、Wronsky行列式、矩阵函数的微分运算、非齐次线性微分方程的解的基本形式、降阶法、常数变易法。

二，直化定理及其结果。

5，存在性与唯一性定理的回顾，Picard逼近。

6，复合映射，变方方程的初值问题与参数，初值问题与参数的解的光滑独立性。

7，直化定理及其结果，首次积分。

2， 初值问题解的连续可微性定理、向量场的方向导数、向量场的李代数、首次积分、Hamilton正则方程组、一阶齐次线性偏微分方程、一阶齐次线性偏微分方程的Cauchy问题。线性与拟线性方程的柯西问题。

9，相流的扭曲。

3， 一阶非齐次线性偏微分方程、一阶拟线性偏微分方程、一阶拟线性偏微分方程的特征线素场、线素场的积分曲面、一阶拟线性偏微分方程解的充要条件、一阶非线性偏微分方程、Hamilton-Jacobi方程、能量的等高线、Hadamard引理、临界与非临界等高线。

4， 微分方程的幂级数解、孤立奇点、Euler方程。

5， 正则奇点、Frobenius方法。

6， Sturm比较定理、边值条件的分类、Sturm边值问题、齐次线性方程的基解、Green函数、线性与非线性边值问题、边值问题解的存在性与唯一性定理。

7， 边值问题Green函数的唯一性定理、含参数的边值问题、Sturm-Liouville

特征值问题、Sturm分离定理、特征值比较定理、振幅定理。

8， 小摄动、保守系统的稳定性、自振、可微等价、拓扑等价、拓扑分类定理、Lyapunov函数。

9， 平面上微分方程的稳定性、导数的估计、Lyapunov稳定性、渐进稳定、特征值与稳定性的关系。

10， 环面上方程的相曲线、Louville定理、周期系数微分方程、周期解、强稳定系统。

三，稳定性与相平面。

10，映射的不动点与微分方程奇点的稳定性。

11，相平面，相平面的拓扑，Poincare映射、稳定性定理、Gronwall引理、非线性方程的不稳定性、Grobman-Hartman定理、指数稳定、Lyapunov定理、极限点、极限集、极限环，不变集。Floquet定理。

12， 吸引子、Chetaev不稳定定理、流、流盒、平面动力系统。

四，确定性与混沌。

12，小振动，环面的密集缠绕，KAM定理。

13，Poincare-Bendixson定理。Poincare-Bendixeon定理、极限环、微分方程解的无限延拓、光滑映射的不动点定理、奇点的指数。

14，Smale马鞍型，符号动力系统初步。

Ordinary Differential Equations & Vector Space

1. V. I. Arnold,  Ordinary Differential Equations, Springer-Verlag, Berlin, 2006.

Coddington & Levinson, “Theory of Ordinary Differnetial Equations” 

2，V.I.Arnold，” Geometrical Methods in the Theory of Ordinary Differential Equations” 。

3，M.W.Hirsch、S.Smale、R.Devaney，Differential Equations，Dynamical Systems，and an Introduction to Chaos，Elsevier，2004。

4，M.V.Fedoryuk，常微分方程，科学出版社，2004。

5，L.S.Pontryagin，常微分方程，科学出版社，1982。

6，B.P.Demidovich，稳定性的数学理论讲义，科学出版社，1967。

7，A.F.Filippov，常微分方程习题集，“混沌与正则动力学”出版社，2000。

彼得罗夫斯基 “常微分方程讲义” 

《高等数学例题与习题集常微分方程》A.K.博亚尔丘克等

卡姆克(Kamke) 常微分方程手册,

《常微分方程基础理论》Po-Fang Hsieh，Yasutaka Sibuya

W.D. Boyce and R.C. DiPrima, Elementary Differential Equations, Wiley, 2009

丁同仁、李承治《常微分方程教程》高等教育出版社

袁相碗《常微分方程》南京大学出版社

《常微分方程（第三版）》王高雄，周之铭，朱思铭，王寿松

《常微分方程学习辅导与习题解答》朱思铭

《常微分方程讲义》叶彦谦

《常微分方程讲义》王柔怀，伍卓群

《常微分方程》伍卓群，李勇

《常微分方程》东北师范大学数学系微分方程教研室

《常微分方程学习指导书》王克，潘家齐

《常微分方程简明教程》曹之江

《常微分方程》方道元

《常微分方程》张伟年

《常微分方程》肖淑贤

《常微分方程习题解》庄万

100《微分方程的理论及其解法》钱伟长著

【习题集】

【辅导书】

107《常微分方程内容、方法与技巧》孙清华, 李金兰, 孙昊著

【提高】

110《常微分方程手册》卡姆克(Kamke)编

作者还著有《一阶偏微分方程手册》、《勒贝格-斯蒂尔吉斯积分》。

111《Handbook of exact solutions for ODEs》(《常微分方程精确解手册》英文版) Polyanin,Zaitsev编著

112《常微分方程补充教程》尤秉礼编

113《常微分方程专题研究》汤光宋著

Differential equations -1

     I put the equation into two categories: functional equation (this “number” is more than real numbers, can also be complex numbers, matrices, and tensors, quaternions, etc), logical equations (that is, non-traditional type of equations). Function equation can be broken down into algebraic, transcendental equations, matrices, differential equations (integral) equations, differential equations, differential equations with difference, usually functional equation (, including iteration). We all know that algebraic equations in five or more no general formula in the form of solutions, basic only numerical solutions of transcendental equations, matrix equations and “number” equation, and usually functions in addition to some skills, most can only be solved by series method. The last differential (integral) equations are not very optimistic, not all solutions of integrable (but the vast majority are not solvable).

     For ordinary differential equations in differential equations, this lesson focuses on are some of the common types of integrable solutions, solutions for qualitative method, numerical solution of mathematical transform, series solution, solution, application of differential equations in geometry and physics.

One, basic concepts and elementary integrals.

1 , Basic concept of differential equation, phase space, the integral curves, differential equations with one dimensional phase space.

2 , With multi-dimensional phase space of differential equations, curves, the successor function,Poincare map, little vibration, existence of solutions with uniqueness andLipscitz conditions.

3 , Variables separable equations, Cartesian product of the two systems. Lotka-Volterra model, the equilibrium position, a first-order linear homogeneous equations, its symmetry group. First order linear homogeneous equations with periodic coefficients.

3 First-order linear equations, single-valued transform coefficients of periodic solutions of linear equations with the cycle.

4 , First order linear non-homogeneous equation, superposition principle,Green function, with periodic coefficients of first order linear non-homogeneous equation, the one-parameter group of diffeomorphisms, vector field, phase.

5 , Diffeomorphisms on the limit cycle, phase current, homogeneous equations, to be homogeneous equations.

4 Fully differential equations, one degree of freedom Hamilton Equation of pendulum.

6 , Elementary integration method and singular solution. Bernoulli equation,Riccati equation, appropriate equations, potential functions, integrating factors, phase planes, picture rail.

Second, the existence theorem.

7 , Generalized Lotka-Volterra model, regular lines, extraordinary solutions, envelope, andClairaut equations,d ‘ Alembertequation,Banach Space, successive approximation, contraction mapping principle.

8 , Vector integration and differentiability and Lipscitz conditions, proof of existence and uniqueness theorem,Peanoexistence theorem, equicontinuous,Ascoli-Arzela theorem,Euler Line method.

7 , The existence and uniqueness of solutions for initial value problems with continuous independence theorem Picard Method.

8 And solutions Picard Approximation method.

9 , Successive approximation of the divergence, well-posed problems, solutions of initial value problem for continuous and differentiable dependence theorem, continuous and differentiable dependence parameter theorem, direct extension theorem, vector field.

10 , Higher order differential equations of first order differential equations, higher order differential equations of existence and uniqueness, the differential equations of higher order differentiability and extensions extension theorem, a differential equation phase space dimension, variational equations, autonomous system, contact structures.

9 Integral curves and curve continuity theorem and its application in linear equations.

11 , Closed curve, the one-parameter group of linear operators, the fundamental theorem of linear equations with constant coefficients, and operator

Determinants, operator of signs and Liouville Formulas, Liouville-Ostrogradskii Formula Diagonalizable operators, characteristic equation, there is friction pendulum equation of the curve.

Third, any first-order linear equations with constant coefficients.

12 , Complex phase space of linear differential equations and the classification of singularities, the characteristic equation has the general solution of linear equations with a single.

11 , Homogeneous equation and the right half of the equation of a plane.

13 , Jordan standard solution of constant coefficient linear differential equations, linear differential equations solution space, solution of non-homogeneous linear differential equations, complex amplitude method and resonance.

Differential equations -2

One, linear systems of equations.

1 , Flow, and linear operators.

2 , Of the complex and real, value calculations.

3 , Plural.

4 , Jordan Block.

1 , Variable coefficient homogeneous linear differential equations, solution of homogeneous linear differential equations with variable coefficients prior estimates, solution of homogeneous linear differential equations with variable coefficients space,Wronsky determinants, matrices, the differential of the function operation, solution of non-homogeneous linear differential equation of the basic form, reduced-order method, the method of variation of constants.

Second, the theorem and its consequences.

5 , Review of existence and Uniqueness theorems, Picard Approximation.

6 Complex mapping, variable equation’s initial value problem and parametric, independence of the smooth solution of the initial value problem and parametric.

7 , Theorems and results of first integrals.

2 , Solutions of initial value problem of continuous differentiability theorem, the directional derivative of a vector field, the lie algebra of vector fields and the first points,Hamilton canonical equation group, first-order homogeneous linear partial differential equations, first order linear homogeneous partial differential equationsCauchy problem. Linear and quasilinear Cauchy problem.

9 , Phase distortion.

3 , First-order non-homogeneous linear first-order quasilinear partial differential equations, partial differential equations, first order quasilinear partial differential equations of the characteristic points of the line element field, the line element field surfaces, the first necessary and sufficient conditions of solutions to quasilinear partial differential equations, first order nonlinear partial differential equations,Hamilton-Jacobi equation, the energy contour,Hadamard Lemma, critical and non-critical contours.

4 , Differential equation by power series solutions, isolated singularities,Euler equations.

5 , Regular singular point,Frobenius method.

6 , Sturm comparison theorem and boundary conditions of classification,Sturm base of homogeneous linear equations with boundary value problems, solutions,Green function, linear and non-linear boundary value problems, boundary value problem for the existence and uniqueness theorem.

7 , Boundary value problems of Green function uniqueness theorem, with a parameter of boundary value problems,Sturm-Liouville

Eigenvalue problem, Sturm Separation theorem, theorem of eigenvalue comparison theorem and amplitude.

8 , Small perturbations, system stability, vibration, differentiable equivalence, topological equivalence, topological classification theorem,Lyapunov functions.

9 , Plane stability of differential equations, derivatives of the estimate and theLyapunov stability and asymptotic stability, characteristic value and the stability of the relationship.

10 , Phase of the toroidal drive above curve,Louville theorem, periodic, periodic solutions, strong stability of differential equation systems.

Third, stability and phase plane.

10 , Map fixed points and stability of differential equations with singularities.

11 , The phase plane, plane topology, Poincare Maps, stability, Gronwall Instability, lemmas, the nonlinear equation Grobman-Hartman Theorems, exponential stability, Lyapunov Limit theorem, limit point, set, Limit cycle, Invariant set. Floquettheorem.

12 , Attracting children,Chetaev instability theorem, flow, flow boxes, plane power systems.

Four, certainty and chaos.

12 , Little vibration, torus intensive wound, KAM Theorem.

13 , Poincare-Bendixson Theorem. Poincare-Bendixeon theorem, limit cycles and differential equation with infinite extension, smooth mapping fixed point theorems, singularities of the index.

14 , Smale Saddle, symbolic dynamical system.

Ordinary Differential Equations & Vector Space

1. V. I. Arnold, Ordinary Differential Equations, Springer-Verlag, Berlin, 2006.

Coddington & Levinson, “Theory of Ordinary Differnetial Equations”

2 , V.I.Arnold , ” Geometrical Methods in the Theory of Ordinary Differential Equations ” 。

3 , M.W.Hirsch 、 S.Smale 、 R.Devaney , Differential Equations , Dynamical Systems , and an Introduction to Chaos , Elsevier , 2004。

4 , M.V.Fedoryuk , Ordinary differential equations, science press, 2004 。

5 , L.S.Pontryagin , Ordinary differential equations, science press, 1982 。

6 , B.P.Demidovich , Mathematical theory of the stability of lectures, science press, 1967 。

7 , A.F.Filippov , Sets of ordinary differential equations, ” Canonical dynamics and chaos ” Publishing House, 2000 。

Petrovsky ” Lectures on ordinary differential equations”

The examples in higher mathematics and problem sets of ordinary differential equations A.K. Boyarchuk

Camden grams (Kamke) Handbook of differential equations,

Of the basic theory of ordinary differential equations Po-Fang Hsieh , Yasutaka Sibuya

W.D. Boyce and R.C. DiPrima, Elementary Differential Equations, Wiley, 2009

D colleagues, Li Chengzhi course in ordinary differential equations, higher education press

Yuan bowl of the ordinary differential equation of the Nanjing University Press

The ordinary differential equations (third edition), Kaohsiung, Wang, and Zhou Zhiming, and Zhu siming, shuosong Wang

The guidance and solutions of ordinary differential equations of Zhu siming

The lectures on ordinary differential equations yanqian ye

The lectures on ordinary differential equations of Wang Rouhuai, Wu zhuoqun

The ordinary differential equation Wu zhuoqun, Li Yong

Of the ordinary differential equation of the Mathematics Department of Northeast Normal University Department of differential equations

Queen of the ordinary differential equations study guide book, Pan Jiaqi

The concise course of ordinary differential equations of Cao Zhi Jiang

Road of the ordinary differential equation

Of the ordinary differential equation Zhang Weinian

Of the ordinary differential equation xiaoshuxian

Of the learning solutions of ordinary differential equations

100 Qian weichang of the theory of differential equations and their solutions a

“Onward”

“Books”

107 Ordinary differential equations of the contents, methods and techniques of Qinghua Sun , Li Jinlan , Sun h a

“Increase”

110 The ODE manual Camden grams (Kamke) Series

The author is also the author of the Handbook of first order partial differential equation, andThe Lebesgue – Stieltjes integral 。

111 《 Handbook of exact solutions for ODEs 》 (English version of the Handbook of exact solutions for ordinary differential equations ) Polyanin, Zaitsev Authoring

112 You Bingli series of the ordinary differential equations added tutorials

113 Tang Guangsong in the study of ordinary differential equations with