Chapter 1 First-order differential equations
1．1 Introduction
1．2 First-order linear differential equations
1．3 The Van Meegeren art forgeries
1．4 Separable equations
1．5 Population models
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Chapter 1 First-order differential equations
1．1 Introduction
1．2 First-order linear differential equations
1．3 The Van Meegeren art forgeries
1．4 Separable equations
1．5 Population models
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 Complex roots
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 of judicious guessing
2．6 Mecharucal vibrations
2．6．1 The Tacoma Bridge disaster
2．6．2 Electrical networks
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 Vector spaces
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
3．7 Linear transformations
3．8 The eigenvalue-eigenvector method of finding solutions
3．9 Complex roots
3．10 Equal roots
3．11 Fundamental matrix solutions； eAt
3．12 The nonhomogeneous equation； variation of parameters
3．13 Solving systems by Laplace transforms
Chapter 4 Qualitative theory of differential equations
4．1 Introduction
4．2 Stability of linear systems
4．3 Stability of equilibrium solutions
4．4 The phase-plane
4．5 Mathematical theories of war
4．5．1 L．F．Richardson's theory of conflict
4．5．2 Lanchester's combat models and the battle of Iwo Jima
4．6 Qualitative properties of orbits
4．7 Phase portraits of linear systems
4．8 Long time behavior of solutions； the Poincare～Bendixson Theorem
4．9 Introduction to bifurcation theory
4．10 Predator-prey problems； or why the percentage of sharks caught in the Mediterranean Sea rose dramatically during World War I
4．11 The principle of competitive exclusion in population biology
4．12 The Threshold Theorem of epidemiology
4．13 A model for the spread of gonorrhea
Chapter 5 Separation of variables and Fourier series
5．1 Two point boundary-value problems
5．2 Introduction to partial differential equations
5．3 The heat equation； separation of variables
5．4 Fourier series
5．5 Even and odd functions
5．6 Return to the heat equation
5．7 The wave equation
5．8 Laplace's equation
Chapter 6 Sturm-Liouville boundary value problems
6．1 Introduction
6．2 Inner product spaces
6．3 Orthogonal bases， Hermitian operators
6．4 Sturm-'Liouville theory
Appendix A
Some simple facts concerning functions of several variables
Appendix B
Sequences and series
Appendix C
C Programs
Answers to odd-numbered exercises
Index
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原文摘录
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Remark: A student of mine once suggested that we use Equation(3)to find the tim t at which p(t)=2, and then we can deduce how long ago mankind appeared on earth. On the surface this seems like a fantastic idea. However, we cannot travel that far backwards into the past, since our model is no longer accurate when the population is small. (查看原文)
To this end, observe that a function y(t) is a solution of (1) if a constant times its second derivative, plus another constant times its first derivative, plus a third constant time itself is identically zero. In other words, the three terms ay", by', and cy must cancel each other. In general, this can only occur if the three functions y(t), y'(t), and y''(t) are of the"same type". (查看原文)
2 有用 7086 2021-01-14 10:30:00
本书对读者水平要求比较低，会计算原函数和不定积分，熟悉导数记号的话，就可阅读。第一章介绍了三两个方程解法后就开始科普微分方程在断案、社会学等方面的应用，这才是学习数学的趣味所在！国内很多数学教材就输在了这一点上：只讲公式，没有方法，应用也是那么几个无聊的例子翻来覆去，无从体现数学的价值。建议尽早来读这本书，因为其理论非常浅显，而应用占到了接近一半的篇幅。但想要了解微分方程的精细理论的同学会失望而返... 本书对读者水平要求比较低，会计算原函数和不定积分，熟悉导数记号的话，就可阅读。第一章介绍了三两个方程解法后就开始科普微分方程在断案、社会学等方面的应用，这才是学习数学的趣味所在！国内很多数学教材就输在了这一点上：只讲公式，没有方法，应用也是那么几个无聊的例子翻来覆去，无从体现数学的价值。建议尽早来读这本书，因为其理论非常浅显，而应用占到了接近一半的篇幅。但想要了解微分方程的精细理论的同学会失望而返。 (展开)