Differential Equations and Linear Algebra

1 First-Order Differential Equations

1.1 Dynamical Systems: Modeling 1

1.2 Solutions and Direction Fields: Qualitative Analysis 11

1.3 Separation of Variables: Quantitative Analysis 25

1.4 Approximation Methods: Numerical Analysis 33

1.5 Picard’s Theorem: Theoretical Analysis 46

 

2 Linearity and Nonlinearity

2.1 Linear Equations: The Nature of Their Solutions 55

2.2 Solving the First-Order Linear Differential Equation 63

2.3 Growth and Decay Phenomena 73

2.4 Linear Models: Mixing and Cooling 80

2.5 Nonlinear Models: Logistic Equation 87

2.6 Systems of Differential Equations: A First Look 100

 

3 Linear Algebra

3.1 Matrices: Sums and Products 115

3.2 Systems of Linear Equations 130

3.3 The Inverse of a Matrix 146

3.4 Determinants and Cramer’s Rule 156

3.5 Vector Spaces and Subspaces 167

3.6 Basis and Dimension 177

 

4 Higher-Order Linear Differential Equations

4.1 The Harmonic Oscillator 195

4.2 Real Characteristic Roots 210

4.3 Complex Characteristic Roots 229

4.4 Undetermined Coefficients 244

4.5 Variation of Parameters 255

4.6 Forced Oscillations 261

4.7 Conservation and Conversion 274

 

5 Linear Transformations

5.1 Linear Transformations 285

5.2 Properties of Linear Transformations 300

5.3 Eigenvalues and Eigenvectors 311

5.4 Coordinates and Diagonalization 327

 

6 Linear Systems of Differential Equations

6.1 Theory of Linear DE Systems 343

6.2 Linear Systems with Real Eigenvalues 357

6.3 Linear Systems with Nonreal Eigenvalues 372

6.4 Stability and Linear Classification 384

6.5 Decoupling a Linear DE System 394

6.6 Matrix Exponential 400

6.7 Nonhomogeneous Linear Systems 410

 

7 Nonlinear Systems of Differential Equations

7.1 Nonlinear Systems 421

7.2 Linearization 431

7.3 Numerical Solutions 441

7.4 Chaos, Strange Attractors, and Period Doubling 449

7.5 Chaos in Forced Nonlinear Systems 456

 

8 Laplace Transforms

8.1 The Laplace Transform and Its Inverse 467

8.2 Solving DEs and IVPs with Laplace Transforms 475

8.3 The Step Function and the Delta Function 485

8.4 The Convolution Integral and the Transfer Function 499

8.5 Laplace Transform Solution of Linear Systems 509

 

9 Discrete Dynamical Systems

9.1 Iterative Equations 517

9.2 Linear Iterative Systems 530

9.3 Nonlinear Iterative Equations: Chaos Again 542

 

10 Control Theory

10.1 Feedback Controls 557

10.2 Introduction to Optimal Control 567

10.3 Pontryagin Maximum Principle 579