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Solving Systems of Differential Equations with Eigenvalues and Eigenvectors
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Classroom Contents
Engineering Math - Differential Equations and Dynamical Systems
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- 1 Differential Equations and Dynamical Systems: Overview
- 2 Calculus Review: The Derivative (and the Power Law and Chain Rule)
- 3 Gentle Introduction to Modeling with Matrices and Vectors: A Probabilistic Weather Model
- 4 The Simplest Ordinary Differential Equation (ODE) and Its Exponential Solution
- 5 Solving Differential Equations with Power Series: A Simple Example
- 6 Taylor Series and Power Series Made Easy (with Pictures): Review of Calculus
- 7 Taylor Series of the Exponential Function and Euler's Formula!
- 8 Second-Order Ordinary Differential Equations: Solving the Harmonic Oscillator Four Ways
- 9 Example Second-Order ODE: Spring-Mass-Damper
- 10 More Examples of Second Order Differential Equations
- 11 High-Order Ordinary Differential Equations with More Derivatives (from Physics)
- 12 Solving General High-Order, Linear Ordinary Differential Equations (ODEs)
- 13 Matrix Systems of Differential Equations
- 14 Motivating Eigenvalues and Eigenvectors with Differential Equations
- 15 Eigenvalues and Eigenvectors
- 16 Solving Systems of Differential Equations with Eigenvalues and Eigenvectors
- 17 2x2 Systems of ODEs: Sources and Sinks
- 18 2x2 Systems of ODEs: Saddle Points and Instability
- 19 2x2 Systems of ODEs: Imaginary Eigenvalues and Center Fixed Points
- 20 Stability and Eigenvalues: What does it mean to be a "stable" eigenvalue?
- 21 What is a "Linear" Differential Equation?
- 22 Linearizing Nonlinear Differential Equations Near a Fixed Point
- 23 A Particle in a Potential Well: Nonlinear Dynamics
- 24 Drawing Phase Portraits for Nonlinear Systems
- 25 Phase Portrait for Double Well Potential
- 26 The Hartman-Grobman Theorem, Structural Stability of Linearization, and Stable/Unstable Manifolds
- 27 Non-Normal Linear Systems and Transient Energy Growth: Bypass Transition to Turbulence
- 28 Repeated Eigenvalues and Secular Terms: Transient Growth in Non-Normal Systems
- 29 Systems of Differential Equations: Diagonalization and Jordan Canonical Form
- 30 Differential Equations with Forcing: Method of Undetermined Coefficients
- 31 Differential Equations with Forcing: Method of Variation of Parameters
- 32 Systems of Differential Equations with Forcing: Example in Control Theory
- 33 Linear Systems of Differential Equations with Forcing: Convolution and the Dirac Delta Function
- 34 Forced Systems of Differential Equations in Matlab and Python
- 35 Numerical Differentiation with Finite Difference Derivatives
- 36 Numerical Differentiation: Second Derivatives and Differentiating Data
- 37 Why we can't take "dt" to 0 in a computer: Sources of error in numerical differentiation
- 38 Numerical Integration: Discrete Riemann Integrals and Trapezoid Rule
- 39 Numerical Simulation of Ordinary Differential Equations: Integrating ODEs
- 40 Deriving Forward Euler and Backward/Implicit Euler Integration Schemes for Differential Equations
- 41 Numerical Integration of ODEs with Forward Euler and Backward Euler in Python and Matlab
- 42 Error Analysis of Euler Integration Scheme for Differential Equations Using Taylor Series
- 43 Stability of Forward Euler and Backward Euler Integration Schemes for Differential Equations
- 44 Runge-Kutta Integrator Overview: All Purpose Numerical Integration of Differential Equations
- 45 Coding a Fourth-Order Runge-Kutta Integrator in Python and Matlab
- 46 Numerical Integration of Chaotic Dynamics: Uncertainty Propagation & Vectorized Integration
- 47 Chaotic Dynamical Systems
- 48 Engineering Math Pre-Req: Quick and Dirty Introduction to Python
- 49 Engineering Math Pre-Req: Quick and Dirty Introduction to Matlab
