Differential Equations
Differential Equations. Instructor: Prof. William Kinney, Math and Computer Science, Bethel University. This course discusses analytic solution methods for ordinary differential equations, including special methods for first- and second-order systems, and transformation methods. Also it provides an analysis of systems of differential equations using linear algebra and qualitative phase plane techniques.
| Lecture 01 - Mathematica and Population Modeling (Exponential Model) |
| Lecture 02 - Exponential and Logistic Growth, Separation of Variables |
| Lecture 03 - Logistic Model Solution Checking, Mixing Problem, Intro to Slope Fields |
| Lecture 04 - Mathematica HW, Slope Fields and Euler's Method |
| Lecture 05 - Newton's Law of Cooling/Heating, Euler's Method, Existence and Uniqueness |
| Lecture 06 - Slope Fields, Existence and Uniqueness, Phase Lines |
| Lecture 07 - Implicit Solutions, Logistic Model w/ Harvesting, Bifurcations, Existence and Uniqueness |
| Lecture 08 - Bifurcations, Linear Equations, Undetermined Coefficients, and Flows |
| Lecture 09 - Bifurcations, Undetermined Coefficients, Integrating Factors, Flows and Flow Maps |
| Lecture 10 - Linearity Proofs, Idea of Integrating Factors, More on Flows |
| Lecture 11 - Harmonic Oscillator, Predator/Prey Model, Review of Exam 1 |
| Lecture 12 - Predator/Prey Model, Vector Fields and Direction Fields |
| Lecture 13 - Interacting Species, Damped Harmonic Oscillator, and Decoupled Systems |
| Lecture 14 - NDSolveValue vs NDSolve, Locator, Euler's Method in 2D, Existence/Uniqueness |
| Lecture 15 - Real-Life Meaning of Solutions, Van der Pol Eq, SIR Model, Lorenz Attractor |
| Lecture 16 - Nullclines, Forced Van der Pol, Lorenz (Sensitive Dependence), Linear Systems |
| Lecture 17 - Linear Systems, Straight Line Solutions, Eigenvalues and Eigenvectors |
| Lecture 18 - Linearity Proof Outlines, Sink Example (Eigenvalues, etc), Intro to Matrix Exponential |
| Lecture 19 - The Matrix Exponential, Complex Eigenvalues Example, Flows of Linear Systems |
| Lecture 20 - Linearity Proof, Symmetric Matrix e-values, Flows and Matrix Exp, Review |
| Lecture 21 - Using Eigenvalues to Classify Simple Harmonic Oscillators |
| Lecture 22 - Zero as an Eigenvalue, Bifurcations of Linear Systems, Trace-Determinant Plane |
| Lecture 23 - Repeated Eigenvalues, Trace-Determinant Plane, 3D Systems, Forced Harmonic Oscillators |
| Lecture 24 - Forced Harmonic Oscillators (and Complexification), Trace-Det Plane, 3D Prob |
| Lecture 25 - Beating Oscillations, Amplitude and Phase for Forced Damped Harmonic Oscillators |
| Lecture 26 - Forced Harmonic Oscillators, Intro to Linearization of Nonlinear Systems |
| Lecture 27 - A Simple Model of Suspension Bridges, Linearization of a Predator-Prey Model |
| Lecture 28 - Linearization, Jacobian Matrices, Separatrices, Review |
| Lecture 29 - Phase Plane for a Complicated Competing Species Model |
| Lecture 30 - Hyperbolic Equilibria, Hamiltonian Systems, Pendulum |
| Lecture 31 - Hyperbolicity, Stability, Hamiltonian and Lyapunov Functions |
| Lecture 32 - Prove a Saddle Point is Unstable, Hamiltonian and Lyapunov Functions, Trapping Regions |
| Lecture 33 - Gradient and Hamiltonian Systems, Trapping Regions, Poincare-Bendixson, Laplace Transform |
| Lecture 34 - Laplace Transform, Heaviside (Unit Step) Function, Shift on t Axis |
| Lecture 35 - Laplace Transform for a Forced Harmonic Oscillator Equation, Trapping Regions |
| Lecture 36 - Nonlinear Bifurcation Problem, Trapping Regions and Poincare-Bendixson, Chaos |
| Lecture 37 - Final Exam Review, Complexification, Hamiltonian Systems, Lyapunov Functions |