本书的引进是为了更好地借鉴国外微积分教学与研究的成功经验，促进我国数学教育与研究事业的发展，提高高等学校数学教育教学质量，为本科高年级和研究生开展常微分程研究工作提供必要的理论依据，本书为原版影印，既可供本科高年级和研究生自学参考，也可做为教材使用。

本书为开展常微分方程研究工作的读者提供必要的准备知识，可作为本科高年级和研究生常微分方程课程教材。

本书内容分为四部分：第一部分(第一、二、三章)的内容包括解的存在性、唯一性、对数据的光滑依赖性，以及解的非唯一性；第二部分(第四、六、七章)讨论线性常微分方程，书中用矩阵的S-N分解代替Jordan分解，前者的计算较后者更容易；第三部分(第八、九、十章)讨论非线性常微分方程的稳定性、渐近稳定性等几何理论；第四部分(第五、十一，十二、十三章)讨论常微分方程的幂级数解，包括线性常微分方程的奇点分类及非线性常微分方程当参数或自变量趋向某奇点时的渐近解等。

Preface

Chapter Ⅰ.Fundamental Theorems of Ordinary Differential Equations

Ⅰ-1.Existence and uniqueness with the Lipschitz condition

Ⅰ-2.Existence without the Lipschitz condition

Ⅰ-3.Some global properties of solutions

Ⅰ-4.Analytic differential equations

Exercises Ⅰ

ChapterⅡ.Dependence on Data

Ⅱ-1.Continuity with respect to initial data and parameters

Ⅱ-2.Differentiability

Exercises Ⅱ

Chapter Ⅲ.Nonuniqueness

Ⅲ-1.Examples

Ⅲ-2.The Kneser theorem

Ⅲ-3.Solution curves on the boundary of R(A)

Ⅲ-4.Maximal and minimal solutions

Ⅲ-5.A comparison theorem

Ⅲ-6.Sufficient conditions for uniqueness

Exercises Ⅲ

Chapter Ⅳ.General Theory of Linear Systems

Ⅳ-1.Some basic results concerning matrices

Ⅳ-2.Homogeneous systems of linear differential equations

Ⅳ-3.Homogeneous systems with constant coefficients

Ⅳ-4.Systems with periodic coefficients

Ⅳ-5.Linear Hamiltonian systems with periodic coefficients

Ⅳ-6.Nonhomogeneous equations

Ⅳ-7.Higher-order scalar equations

Exercises Ⅳ

Chapter Ⅴ.Singularities of the First Kind

Ⅴ-1.Formal solutions of an algebraic differential equation

Ⅴ-2.Convergence of formal solutions of a system of the first kind

Ⅴ-3.The S-N decomposition of a matrix of infinite order

Ⅴ-4.The S-N decomposition of a differential operator

Ⅴ-5.A normal form of a differential operator

Ⅴ-6.Calculation of the normal form of a differential operator

Ⅴ-7.Classification of singularities of homogeneous linear systems

Exercises Ⅴ

Chapter Ⅵ.Boundary-Value Problems of Linear Differential Equations of the Second-Order

Ⅵ-1.Zeros of solutions

Ⅵ-2.Sturm-Liouville problems

Ⅵ-3.Eigenvalue problems

Ⅵ-4.Eigenfunction expansions

Ⅵ-5.Jost solutions

Ⅵ-6.Scattering data

Ⅵ-7.Refiectionless potentials

Ⅵ-8.Construction of a potential for given data

Ⅵ-9.Differential equations satisfied by reflectionless potentials

Ⅵ-10.Periodic potentials

Exercises Ⅵ

Chapter Ⅶ.Asymptotic Behavior of Solutions of Linear Systems

Ⅶ-1.Liapounoff's type numbers

Ⅶ-2.Liapounoff's type numbers of a homogeneous linear system

Ⅶ-3.Calculation of Liapounoff's type numbers of solutions

Ⅶ-4.A diagonalization theorem

Ⅶ-5.Systems with asymptotically constant coefficients

Ⅶ-6.An application of the Floquet theorem

Exercises Ⅶ

Chapter Ⅷ.Stability

Ⅷ-1.Basic definitions

Ⅷ-2.A sufficient condition for asymptotic stability

Ⅷ-3.Stable manifolds

Ⅷ-4.Analytic structure of stable manifolds

Ⅷ-5.Two-dimensional linear systems with constant coefficients

Ⅷ-6.Analytic systems in R2

Ⅷ-7.Perturbations of an improper node and a saddle point

Ⅷ-8.Perturbations of a proper node

Ⅷ-9.Perturbation of a spiral point

Ⅷ-10.Perturbation of a center

Exercises Ⅷ

Chapter Ⅸ.Autonomous Systems

Ⅸ-1.Limit-invariant sets

Ⅸ-2.Liapounoff's direct method

Ⅸ-3.Orbital stability

Ⅸ-4.The Poincare-Bendixson theorem

Ⅸ-5.Indices of Jordan curves

Exercises Ⅸ

Chapter Ⅹ.The Second-Order Differential Equation (d2x)/(dt2)+h(x)*(dx)/(dt)+g(x)=0

Ⅹ-1.Two-point boundary-value problems

Ⅹ-2.Applications of the Liapounoff functions

Ⅹ-3.Existence and uniqueness of periodic orbits

Ⅹ-4.Multipliers of the periodic orbit of the van der Pol equation

Ⅹ-5.The van der Pol equation for a small ε > 0

Ⅹ-6.The van der Pol equation for a large parameter

Ⅹ-7.A theorem due to M.Nagumo

Ⅹ-8.A singular perturbation problem

Exercises Ⅹ

Chapter Ⅺ.Asymptotic Expansions

Ⅺ-1.Asymptotic expansions in the sense of Poincare

Ⅺ-2.Gevrey asymptotics

Ⅺ-3.Flat functions in the Gevrey asymptotics

Ⅺ-4.Basic properties of Gevrey asymptotic expansions

Ⅺ-5.Proof of Lemma Ⅺ-2-6

Exercises Ⅺ

Chapter Ⅻ.Asymptotic Expansions in a Parameter

Ⅻ-1.An existence theorem

Ⅻ-2.Basic estimates

Ⅻ-3.Proof of Theorem Ⅻ-1-2

Ⅻ-4.A block-diagonalization theorem

Ⅻ-5.Gevrey asymptotic solutions in a parameter

Ⅻ-6.Analytic simplification in a parameter

Exercises Ⅻ

Chapter ⅩⅢ.Singularities of the Second Kind

ⅩⅢ-1.An existence theorem

ⅩⅢ-2.Basic estimates

ⅩⅢ-3.Proof of Theorem ⅩⅢ-1-2

ⅩⅢ-4.A block-diagonalization theorem

ⅩⅢ-5.Cyclic vectors (A lemma of P.Deligne)

ⅩⅢ-6.The Hukuhara-Turrittin theorem

ⅩⅢ-7.An n-th-order linear differential equation at a singular point of the second kind

ⅩⅢ-8.Gevrey property of asymptotic solutions at an irregular singular point

Exercises ⅩⅢ

References

Index