Elementary Differential Equations
暫譯: 初等微分方程

Werner E. Kohler, Lee W. Johnson

Elementary Differential Equations
  • 出版商: Addison Wesley
  • 出版日期: 2002-12-06
  • 售價: $1,029
  • 語言: 英文
  • 頁數: 760
  • 裝訂: Hardcover
  • ISBN: 0201709260
  • ISBN-13: 9780201709261

    • 已絕版

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商品描述

Elementary Differential Equations integrates the underlying theory, the solution procedures, and the numerical/computational aspects of differential equations in a seamless way. For example, whenever a new type of problem is introduced (such as first-order equations, higher-order equations, systems of differential equations, etc.) the text begins with the basic existence-uniqueness theory. This provides the student the necessary framework to understand and solve differential equations. Theory is presented as simply as possible with an emphasis on how to use it. The Table of Contents is comprehensive and allows flexibility for instructors.

Table of Contents

1. Introduction to Differential Equations.

Examples of Differential Equations.

Direction Fields.



2. First Order Linear Differential Equations.

Existence and Uniqueness.

First Order Linear Homogeneous Differential Equations.

First Order Linear Nonhomogeneous Differential Equations.

Introduction to Mathematical Models.

Mixing Problems and Cooling Problems.



3. First Order Nonlinear Differential Equations.

Existence and Uniqueness.

Separable First Order Equations.

Exact Differential Equations.

Bernoulli Equations.

The Logistic Population Model.

One-Dimensional Motion with Air Resistance.

One-Dimensional Dynamics with Distance as the Independent Variable.

Euler's Method.



4. Second Order Linear Differential Equations.

Existence and Uniqueness.

The General Solution of Homogeneous Equations.

Fundamental Sets and Linear Independence.

Constant Coefficient Homogeneous Equations.

Real Repeated Roots; Reduction of Order.

Complex Roots.

Unforced Mechanical Vibrations.

The General Solution of the Linear Nonhomogeneous Equation.

The Method of Undetermined Coefficients.

The Method of Variation of Parameters.

Forced Mechanical Vibrations, Electrical Networks, and Resonance.



5. Higher Order Linear Differential Equations.

Existence and Uniqueness.

The General Solution of nth Order Linear Homogeneous Equation.

Fundamental Sets and Linear Independence.

Constant Coefficient Homogeneous Equations.

Nonhomogeneous Linear Equations.



6. First Order Linear Systems.

The Calculus of Matrix Functions.

Existence and Uniqueness.

Homogeneous Linear Systems.

Fundamental Sets and Linear Independence.

Constant Coefficient Homogeneous Systems.

Complex Eigenvalues.

Repeated Eigenvalues.

Nonhomogeneous Linear Systems.

Euler's Method for Systems of Differential Equations.

Diagonalization.

Propagator Matrices, Functions of a Matrix and the Exponential Matrix.



7. Laplace Transforms.

The Laplace Transform.

Laplace Transform Pairs.

Review of Partial Fractions.

Solving Scalar Problems. Laplace Transforms of Periodic Functions.

Solving Systems of Differential Equations.

Convolution.

The Delta Function and Impulse Response.



8. Nonlinear Systems.

Existence and Uniqueness.

Equilibrium Solutions and Direction Fields.

Conservative Systems.

Stability.

Linearization and the Local Picture.

The Two-dimensional Linear System y1=Ay.

Predator-Prey Population Models.



9. Numerical Methods.

Introduction.

Euler's Method, Heun's Method, the Modified Euler's Method.

Taylor Series Methods.

Runge-Kutta Methods.



10. Series Solution of Differential Equations.

Review of Power Series.

Series Solutions near an Ordinary Point.

The Euler Equation.

Solutions Near a Regular Singular Point; the Method of Frobenius.

The Method of Frobenius Continued; Special Cases and a Summary.

商品描述(中文翻譯)

《初等微分方程》將微分方程的基本理論、解題程序以及數值/計算方面無縫整合。例如，當引入一種新類型的問題（如一階方程、高階方程、微分方程系統等）時，文本會從基本的存在性-唯一性理論開始。這為學生提供了理解和解決微分方程所需的框架。理論以盡可能簡單的方式呈現，並強調如何使用它。目錄內容全面，並為教學者提供靈活性。

目錄

1. 微分方程簡介。

微分方程的例子。

方向場。

2. 一階線性微分方程。

存在性和唯一性。

一階線性齊次微分方程。

一階線性非齊次微分方程。

數學模型簡介。

混合問題和冷卻問題。

3. 一階非線性微分方程。

存在性和唯一性。

可分離的一階方程。

精確微分方程。

伯努利方程。

邏輯斯蒂人口模型。

帶空氣阻力的一維運動。

以距離為自變量的一維動力學。

歐拉法。

4. 二階線性微分方程。

存在性和唯一性。

齊次方程的一般解。

基本解集和線性獨立性。

常數係數齊次方程。

實重根；降階。

複重根。

無外力機械振動。

線性非齊次方程的一般解。

不定係數法。

參數變化法。

強迫機械振動，電路網絡和共振。

5. 高階線性微分方程。

存在性和唯一性。

n階線性齊次方程的一般解。

基本解集和線性獨立性。

常數係數齊次方程。

非齊次線性方程。

6. 一階線性系統。

矩陣函數的微積分。

存在性和唯一性。

齊次線性系統。

基本解集和線性獨立性。

常數係數齊次系統。

複特徵值。

重特徵值。

非齊次線性系統。

微分方程系統的歐拉法。

對角化。

傳播矩陣、矩陣函數和指數矩陣。

7. 拉普拉斯變換。

拉普拉斯變換。

拉普拉斯變換對。

部分分式回顧。

解決標量問題。週期函數的拉普拉斯變換。

解決微分方程系統。

卷積。

德爾塔函數和脈衝響應。

8. 非線性系統。

存在性和唯一性。

平衡解和方向場。

保守系統。

穩定性。

線性化和局部圖像。

二維線性系統 y1=Ay。

捕食者-獵物人口模型。

9. 數值方法。

簡介。

歐拉法、海因法、改進的歐拉法。

泰勒級數方法。

龍格-庫塔方法。

10. 微分方程的級數解。

冪級數回顧。

在常規點附近的級數解。

歐拉方程。

在正則奇點附近的解；弗羅貝尼烏斯法。

弗羅貝尼烏斯法的延續；特例和總結。

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