本书以具有基础微积分知识的大学本科学生为对象，全面地介绍了微分方程及其应用，体例清晰，可作为一个或者二个学期的教程。该教材区别于其他微积分教材的显著特点是，涉及微积分在诸多领域的应用和*的研究进展。其应用主要有：证明Rembrandt Society of Belgium 购买的画作Disciples at Emmaus是赝品：对塔科马桥梁灾难的数学解释；达尔文定理的证明。分支理论、C、Pascal和 Fortran等计算机程序的介绍以及许多原始的且具有挑战性的习题进一步提高了该书的价值。 目次：一阶微分方程；二阶线性微分方程；微分方程体系；微分方程定性理论；变量分离和傅里叶级数；Sturm-Liouville边界值问题；附录；索引。 读者对象：数学专业科研人员及学生。
目录 Chapter 1 First-order differential equations 1.1 Introduction 1.2 First-orderlinear differential equations 1.3 The Van Meegeren art forgeries 1.4 Separableequations 1.5 Populationmodels 1.6 The spread of technological innovations 1.7 An atomic waste disposal problem 1.8 The dynamics of tumor growth， nuxing problems， and orthogonal trajectories 1.9 Exact equations， and why we cannot solve very many differential equations 1.10 The existence-uniqueness theorem； Picard iteration 1.11 Finding roots of equations by iteration 1.11.1 Newton's method 1.12 Difference equations， and how to compute the interest due on your student loans 1.13 Numerical approximations； Euler's method 1.13.1 Error analysis for Euler's method 1.14 The three term Taylor series method 1.15 An improved Euler method 1.16 The Runge-Kutta method 1.17 What to do in practice Chapter 2 Second-order linear differential equations 2.1 Algebraic properties of solutions 2.2 Linear equations with constant coefficients 2.2.1 Complexroots 2.2.2 Equal roots； reduction of order 2.3 The nonhomogeneous equation 2.4 The method of variation of parameters 2.5 The method ofjudicious guessing 2.6 Mecharucalvibrations 2.6.1 The Tacoma Bridge disaster 2.6.2 Electricalnetworks 2.7 A model for the detection of diabetes 2.8 Series solutions 2.8.1 Singular points， Euler equations 2.8.2 Regular singular points， the method of Frobenius 2.8.3 Equal roots， and roots differing by an integer 2.9 The method of Laplace transforms 2.10 Some useful properties of Laplace transforms 2.11 Differential equations with discontinuous right-hand sides 2.12 The Dirac delta function 2.13 The convolution integral 2.14 The method of elimination for systems 2.15 Higher-order equations Chapter 3 Systems of differential equations 3.1 Algebraic properties of solutions of linear systems 3.2 Vectorspaces 3.3 Dimension of a vector space 3.4 Applications of linear algebra to differential equations 3.5 The theory of determinants 3.6 Solutions of simultaneous linear equations
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