1. FIRST-ORDER DIFFERENTIAL EQUATIONS.
Modeling via Differential Equations. Analytic Technique: Separation of Variables. Qualitative Technique: Slope Fields. Numerical Technique: Euler's Method. Existence and Uniqueness of Solutions. Equilibria and the Phase Line. Bifurcations. Linear Equations. Integrating Factors for Linear Equations.
2. FIRST-ORDER SYSTEMS.
Modeling via Systems. The Geometry of Systems. Analytic Methods for Special Systems. Euler's Method for Systems. The Lorenz Equations.
3. LINEAR SYSTEMS.
Properties of Linear Systems and the Linearity Principle. Straight-Line Solutions. Phase Planes for Linear Systems with Real Eigenvalues. Complex Eigenvalues. Special Cases: Repeated and Zero Eigenvalues. Second-Order Linear Equations. The Trace-Determinant Plane. Linear Systems in Three Dimensions.
4. FORCING AND RESONANCE.
Forced Harmonic Oscillators. Sinusoidal Forcing. Undamped Forcing and Resonance. Amplitude and Phase of the Steady State. The Tacoma Narrows Bridge.
5. NONLINEAR SYSTEMS.
Equilibrium Point Analysis. Qualitative Analysis. Hamiltonian Systems. Dissipative Systems. Nonlinear Systems in Three Dimensions. Periodic Forcing of Nonlinear Systems and Chaos.
6. LAPLACE TRANSFORMS.
Laplace Transforms. Discontinuous Functions. Second-Order Equations. Delta Functions and Impulse Forcing. Convolutions. The Qualitative Theory of Laplace Transforms.
7. NUMERICAL METHODS.
Numerical Error in Euler's Method. Improving Euler's Method. The Runge-Kutta Method. The Effects of Finite Arithmetic.
8. DISCRETE DYNAMICAL SYSTEMS.
The Discrete Logistic Equation. Fixed Points and Periodic Points. Bifurcations. Chaos. Chaos in the Lorenz System.
A. Changing Variables.
B. The Ultimate Guess.
C. Complex Numbers and Euler's Formula.