Elementary Differential Equations and Boundary Value Problems

Cover
Title Page
Copyright
Dedication
The Authors
Preface
Brief Contents
Contents
CHAPTER 1: Introduction
	1.1. Some Basic Mathematical Models; Direction Fields
		Problems
	1.2. Solutions of Some Differential Equations
		Problems
	1.3. Classification of Differential Equations
		Problems
	References
CHAPTER 2: First-Order Differential Equations
	2.1. Linear Differential Equations; Method of Integrating Factors
		Problems
	2.2. Separable Differential Equations
		Problems
	2.3. Modeling with First-Order Differential Equations
		Problems
	2.4. Differences Between Linear and Nonlinear Differential Equations
		Problems
	2.5 Autonomous Differential Equations and Population Dynamics
		Problems
	2.6. Exact Differential Equations and Integrating Factors
		Problems
	2.7. Numerical Approximations: Euler’s Method
		Problems
	2.8. The Existence and Uniqueness Theorem
		Problems
	2.9. First-Order Difference Equations
		Problems
	Chapter Review Problems
	References
CHAPTER 3: Second-Order Linear Differential Equations
	3.1. Homogeneous Differential Equations with Constant Coefficients
		Problems
	3.2. Solutions of Linear Homogeneous Equations; the Wronskian
		Problems
	3.3. Complex Roots of the Characteristic Equation
		Problems
	3.4. Repeated Roots; Reduction of Order
		Problems
	3.5. Nonhomogeneous Equations; Method of Undetermined Coefficients
		Problems
	3.6. Variation of Parameters
		Problems
	3.7. Mechanical and Electrical Vibrations
		Problems
	3.8. Forced Periodic Vibrations
		Problems
	References
CHAPTER 4: Higher-Order Linear Differential Equations
	4.1. General Theory of nth Order Linear Differential Equations
		Problems
	4.2. Homogeneous Differential Equations with Constant Coefficients
		Problems
	4.3. The Method of Undetermined Coefficients
		Problems
	4.4. The Method of Variation of Parameters
		Problems
	References
CHAPTER 5: Series Solutions of Second-Order Linear Equations
	5.1. Review of Power Series
		Problems
	5.2. Series Solutions Near an Ordinary Point, Part I
		Problems
	5.3. Series Solutions Near an Ordinary Point, Part II
		Problems
	5.4. Euler Equations; Regular Singular Points
		Problems
	5.5. Series Solutions Near a Regular Singular Point, Part I
		Problems
	5.6. Series Solutions Near a Regular Singular Point, Part II
		Problems
	5.7. Bessel’s Equation
		Problems
	References
CHAPTER 6: The Laplace Transform
	6.1. Definition of the Laplace Transform
		Problems
	6.2. Solution of Initial Value Problems
		Problems
	6.3. Step Functions
		Problems
	6.4. Differential Equations with Discontinuous Forcing Functions
		Problems
	6.5. Impulse Functions
		Problems
	6.6. The Convolution Integral
		Problems
	References
CHAPTER 7: Systems of First-Order Linear Equations
	7.1. Introduction
		Problems
	7.2. Matrices
		Problems
	7.3. Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors
		Problems
	7.4. Basic Theory of Systems of First-Order Linear Equations
		Problems
	7.5. Homogeneous Linear Systems with Constant Coefficients
		Problems
	7.6. Complex-Valued Eigenvalues
		Problems
	7.7 Fundamental Matrices
		Problems
	7.8. Repeated Eigenvalues
		Problems
	7.9. Nonhomogeneous Linear Systems
		Problems
	References
CHAPTER 8: Numerical Methods
	8.1. The Euler or Tangent Line Method
		Problems
	8.2. Improvements on the Euler Method
		Problems
	8.3. The Runge-Kutta Method
		Problems
	8.4. Multistep Methods
		Problems
	8.5. Systems of First-Order Equations
		Problems
	8.6. More on Errors; Stability
		Problems
	References
CHAPTER 9: Nonlinear Differential Equations and Stability
	9.1. The Phase Plane: Linear Systems
		Problems
	9.2. Autonomous Systems and Stability
		Problems
	9.3. Locally Linear Systems
		Problems
	9.4. Competing Specie
		Problems
	9.5. Predator - Prey Equations
		Problems
	9.6. Liapunov’s Second Method
		Problems
	9.7. Periodic Solutions and Limit Cycles
		Problems
	9.8. Chaos and Strange Attractors: The Lorenz Equations
		Problems
	References
CHAPTER 10: Partial Differential Equations and Fourier Series
	10.1. Two-Point Boundary Value Problems
		Problems
	10.2. Fourier Series
		Problems
	10.3. The Fourier Convergence Theorem
		Problems
	10.4. Even and Odd Functions
		Problems
	10.5. Separation of Variables; Heat Conduction in a Rod
		Problems
	10.6. Other Heat Conduction Problems
		Problems
	10.7. The Wave Equation: Vibrations of an Elastic String
		Problems
	10.8. Laplace’s Equation
		Problems
	A. APPENDIX
	B. APPENDIX
	References
CHAPTER 11: Boundary Value Problems and Sturm-Liouville Theory
	11.1. The Occurrence of Two-Point Boundary Value Problems
		Problems
	11.2 Sturm-Liouville Boundary Value Problems
		Problems
	11.3. Nonhomogeneous Boundary Value Problems
		Problems
	11.4. Singular Sturm-Liouville Problems
		Problems
	11.5. Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion
		Problems
	11.6. Series of Orthogonal Functions: Mean Convergence
		Problems
	References
Answers to Problems
	Chapter 1
		Section 1.1, page 8
		Section 1.2, page 15
		Section 1.3, page 22
	Chapter 2
		Section 2.1, page 31
		Section 2.2, page 38
		Section 2.3, page 47
		Section 2.4, page 57
		Section 2.5, page 67
		Section 2.6, page 75
		Section 2.7, page 82
		Section 2.8, page 90
		Section 2.9, page 99
		Miscellaneous Problems, page 100
	Chapter 3
		Section 3.1, page 109
		Section 3.2, page 119
		Section 3.3, page 125
		Section 3.4, page 132
		Section 3.5, page 141
		Section 3.6, page 146
		Section 3.7, page 157
		Section 3.8, page 167
	Chapter 4
		Section 4.1, page 173
		Section 4.2, page 180
		Section 4.3, page 184
		Section 4.4, page 188
	Chapter 5
		Section 5.1, page 195
		Section 5.2, page 204
		Section 5.3, page 209
		Section 5.4, page 218
		Section 5.5, page 223
		Section 5.6, page 229
		Section 5.7, page 239
	Chapter 6
		Section 6.1, page 247
		Section 6.2, page 255
		Section 6.3, page 262
		Section 6.4, page 268
		Section 6.5, page 273
		Section 6.6, page 279
	Chapter 7
		Section 7.1, page 284
		Section 7.2, page 293
		Section 7.3, page 303
		Section 7.4, page 308
		Section 7.5, page 318
		Section 7.6, page 327
		Section 7.7, page 336
		Section 7.8, page 343
		Section 7.9, page 351
	Chapter 8
		Section 8.1, page 361
		Section 8.2, page 366
		Section 8.3, page 370
		Section 8.4, page 375
		Section 8.5, page 378
		Section 8.6, page 386
	Chapter 9
		Section 9.1, page 397
		Section 9.2, page 406
		Section 9.3, page 415
		Section 9.4, page 426
		Section 9.5, page 433
		Section 9.7, page 452
		Section 9.8, page 460
	Chapter 10
		Section 10.1, page 468
		Section 10.2, page 476
		Section 10.3, page 481
		Section 10.4, page 487
		Section 10.5, page 495
		Section 10.6, page 502
		Section 10.7, page 512
		Section 10.8, page 520
		Section 11.1, page 533
		Section 11.2, page 543
		Section 11.3, page 553
		Section 11.4, page 561
		Section 11.5, page 564
		Section 11.6, page 571
Index