Differential Equations

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.