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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 37 - Final Exam Review, Complexification, Hamiltonian Systems, Lyapunov Functions

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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