The Authors' goal is to communicate an exciting new approach to Differential Equations - through Modeling, Visualization and Dynamical Systems. This new way of looking at ODEs blends the tried and true analytical methods with mathematical modeling, applications to engineering and the sciences, and geometric visualization via numerical solvers. The resulting rich insight and highly motivated learning offers students a powerful, stimulating, yet accessible experience that brings them to a deep understanding of ODEs!
1: Modeling and Differential Equations.
1.1 The Modeling Approach.
1.2 A Modeling Adventure.
1.3 Models and Initial Value Problems.
1.4 The Modeling Process: Differential Systems.
SPOTLIGHT ON MODELING: RADIOCARBON DATING.
SPOTLIGHT ON MODELING: COLD MEDICATION I.
2: First-Order Differential Equations.
2.1 Linear Differential Equations.
2.2 Linear Differential Equations: Qualitative Analysis.
2.3 Existence and Uniqueness of Solutions.
2.4 Visualizing Solution Curves: Slope Fields.
2.5 Separable Differential Equations: Planar Systems.
2.6 A Predator-Prey Model: the Lotka-Volterra System.
2.7 Extension of Solutions: Long-Term Behavior.
2.8 Qualitative Analysis: State Lines, Sign Analysis.
2.9 Bifucations: A Harvested Logistic Model.
Snapshot on Solution Formula Techniques.
SPOTLIGHT ON APPROXIMATE NUMERICAL SOLUTIONS.
SPOTLIGHT ON COMPUTER IMPLEMENTATION.
SPOTLIGHT ON STEADY STATES: LINEAR ODES.
SPOTLIGHT ON MODELING: COLD MEDICATION II.
SPOTLIGHT ON CHANGE OF VARIABLES: PURSUIT MODELS.
SPOTLIGHT ON CONTINUITY IN THE DATA.
3: Second-Order Differential Equations.
3.1 Models of Springs.
3.2 Undriven Constant-Coefficient Linear Differential Equations.
3.3 Visualizing Graphs of Solutions: Direction Fields.
3.4 Periodic Solutions: Simple Harmonic Motion.
3.5 Driven Linear ODEs: Undetermined Coefficients I.
3.6 Driven Linear ODEs: Undetermined Coefficients II.
3.7 Theory of Second-Order Linear Differential Equations.
3.8 Nonlinear Second-Order Differential Equations.
A Snapshot Look at Constant-Coefficient Polynomial Operators.
SPOTLIGHT ON MODELING: VERTICAL MOTION.
SPOTLIGHT ON MODELING: SHOCK ABSORBERS.
SPOTLIGHT ON EINSTEIN'S FIELD EQUATIONS.
4: Applications of Second-Order Differential Equations.
4.1 Newton's Laws: The Pendulum.
4.2 Beats and Resonance.
4.3 Frequency Response Modeling.
4.4 Electrical Circuits.
Snapshot on Mechanical and Electrical Models.
SPOTLIGHT ON MODELING: TUNING A CIRCUIT.
5: The Laplace Transform.
5.1 The Laplace Transform: Solving IVPs.
5.2 Working with the Transform.
5.3 Transforms of Periodic Functions.
5.4 Convolution.
SPOTLIGHT ON THE DELTA FUNCTION.
SPOTLIGHT ON MODELING: TIME DELAYS AND COLLISIONS.
6: Linear Systems of Differential Equations.
6.1 Compartment Models: Tracking Lead.
6.2 Eigenvalues, Eigenvectors and Eigenspaces of Matrices.
6.3 Undriven Linear Differential Systems: Real Eigenvalues.
6.4 Undriven Linear Systems: Complex Eigenvalues.
6.5 Orbital Portraits for Planar Systems.
6.6 Driven Systems: The Matrix Exponential.
6.7 Steady States.
6.8 The Theory of General Linear Systems.
SPOTLIGHT ON VECTORS, MATRICES, INDEPENDENCE.
SPOTLIGHT ON LINEAR ALGEBRAIC EQUATIONS.
SPOTLIGHT ON BIFURCATIONS: SENSITIVITY.
7: Nonlinear Differential Systems.
7.1 Chemical Kinetics: The Fundamental Theorem.
7.2 Properties of Autonomous Systems, Direction Fields.
7.3 Interacting Species: Cooperation, Competition.
SPOTLIGHT ON MODELING: DESTRUCTIVE COMPETITION.
SPOTLIGHT ON MODELING: BIFURCATION AND SENSITIVITY.
8: Stability.
8.1 Stability and Linear Autonomous Systems.
8.2 Stability and Nonlinear Autonomous Systems.
Stability of PlanarNonlinear Systems.
8.3 Conservative Systems.
SPOTLIGHT ON LYAPUNOV FUNCTIONS.
SPOTLIGHT ON ROTATING BODIES.
9: Nonlinear Systems: Cycles and Chaos.
9.1 Cycles.
9.2 Solution Behavior in Planar Autonomous Systems.
9.3 Bifucations.
9.4 Chaos.
SPOTLIGHT ON CHAOTIC SYSTEMS.
10: Fourier Series and Partial Differential Equations.
10.1 Vibrations of a Guitar String.
10.2 Fourier Trigonometric Series.
10.3 Half-Range and Exponential Fourier Series.
10.4 Temperature in a Thin Rod.
10.5 Sturm-Liouville Problems.
10.6 The Method of Eigenfunction Expansions.
SPOTLIGHT ON DECAY ESTIMATES.
SPOTLIGHT ON THE OPTIMAL DEPTH FOR A WINE CELLAR.
SPOTLIGHT ON APPROXIMATION OF FUNCTIONS.
11: Series Solutions.
11.1 The Method of Power Series.
11.2 Series Solutions Near an Ordinary Point.
11.3 Regular Singular Points: Euler's ODE.
11.4 Series Solutions Near Regular Singular Points.
SPOTLIGHT ON LEGENDRE POLYNOMIALS.
SPOTLIGHT ON BESSEL FUNCTIONS.
Appendix A: Basic Theory of Initial Value Problems.
A.1 Uniqueness.
A.2 The Picard Process for Solving an Initial Value Problem.
A.3 Extention of Solutions.
Appendix B: Background Information.
B.1 Power Series.
B.2 Results from Calculus.
Answers to Selected Problems.
Index.
Web Spotlights.
*27 Additional SPOTLIGHTS appear on the text's Web site at www.wiley.com/college/borrelli.