Chapter 1 Differential Equations

1.1 Definitions and Terms 2
1.2 Initial Value Problem 18
1.3 Differential Equations as Mathematical Models 27
Chapter 1 Review 43

Chapter 2 First-Order Differential Equations

2.1 Unsolved curves 48
2.2 Separable Differential Equations 62
2.3 Linear Equations 75
2.4 Exact Differential Equations 87
2.5 Solution by Substitution 97
2.6 Numerical Methods 103
Chapter 2 Review 109

Chapter 3: Models of First-Order Differential Equations

3.1 Linear Model 114
3.2 Nonlinear Model 130
3.3 Model of a System of First-Order Differential Equations 143
Chapter 3 Review 153

Chapter 4 Higher-Order Differential Equations

4.1 Linear Equation Theory 158
4.2 Reduction of coefficient 174
4.3 Homogeneous linear differential equations with constant coefficients 179
4.4 Undetermined coefficients-overlap approach 189
4.5 Undecided Coefficient - Film Approach 201
4.6 Parameter Variation Method 211
4.7 Cauchy-Euler Equations 219
4.8 Green's function 228
4.9 Solution of Systems of Linear Differential Equations by Elimination 242
4.10 Nonlinear Differential Equations 248
Chapter 4 Review 254

Chapter 5: Modeling Higher-Order Differential Equations

5.1 Linear Models: The Initial Value Problem 258
5.2 Linear Models: The Boundary Value Problem 282
5.3 Nonlinear Models 294
Chapter 5 Review 306

Chapter 6 Series Solutions to Linear Differential Equations

6.1 Power Series Review 314
6.2 Solution near the normal point 322
6.3 Solutions near singularities 334
6.4 Special Functions 3447
Chapter 6 Review 367

Chapter 7 Laplace Transforms

7.1 Definition of the Laplace Transform 370
7.2 Inverse Transforms and Derivative Transforms 381
7.3 Operational Properties I 393
7.4 Operational Properties II 410
7.5 Dirac's Delta Function 427
7.6 Systems of Linear Differential Equations 433
Chapter 7 Review 441

Chapter 8: Systems of First-Order Linear Differential Equations

8.1 Basic Theory - Linear Systems of Equations 446
8.2 Homogeneous Linear Systems of Equations 457
8.3 Inhomogeneous Linear Systems of Equations 476
8.4 Matrix Index 485
Chapter 8 Review 490

Chapter 9 Numerical Solutions of Ordinary Differential Equations

9.1 Euler's Method and Error Analysis 494
9.2 Runge-Kutta Method 501
9.3 Multi-step method 507
9.4 Higher-Order Equations and Systems of Linear Equations 511
9.5 Second-Order Boundary Value Problem 516
Chapter 9 Review 521

Chapter 10: Systems of Nonlinear Differential Equations

10.1 Autonomous Systems of Equations 524
10.2 Stability of Linear Systems of Equations 532
10.3 Linearization and Local Stability 542
10.4 Autonomous Systems of Equations as Mathematical Models 554
Chapter 10 Review 564

Chapter 11 Fourier Series

11.1 Orthogonal Functions 568
11.2 Fourier Series 576
11.3 Fourier Cosines and Sine Series 583
11.4 Sturm-Liouvile Problem 592
11.5 Bessel and Legendre series 603
Chapter 11 Review 612

Chapter 12: Boundary Value Problems in Cartesian Coordinate Systems

12.1 Separable Partial Differential Equations 616
12.2 Classical Partial Differential Equations and Boundary Value Problems 622
12.3 Heat Equation 629
12.4 Wave Equation 634
12.5 Laplace's Equation 642
12.6 Non-homogeneous boundary value problem 649
12.7 Orthogonal Series Expansion 665
12.8 High-Dimensional Problems 665
Chapter 12 Review 669

Chapter 13: Boundary Value Problems in Different Coordinate Systems

13.1 Polar Coordinates 674
13.2 Polar and Cylindrical Coordinates 680
13.3 Spherical Coordinates 691
Chapter 13 Review 695

Chapter 14 Integral Transforms

14.1 Error Function 700
14.2 Laplace Transform 703
14.3 Fourier Integral 713
14.4 Fourier Transform 720
14.5 Finite Fourier Transform 728
Chapter 14 Review 733

Chapter 15 Numerical Solutions of Partial Differential Equations

15.1 Laplace's Equation 738
15.2 Heat Equation 745
15.3 Wave Equation 752
Chapter 15 Review 757

supplement
Appendix A Functions Defined by Integral 760
Appendix B Matrix 770
Appendix C: Laplace Transforms 792

Practice Problem Answers 797
Search 835