Learn · Differential Equations
Lecture 55 of 122Gain and Phase Lag
Stability of Constant Coefficient Second Order DE's
Prof. Arthur Mattuck and others · Massachusetts Institute of Technology
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- From
- Differential Equations, Massachusetts Institute of Technology (at MIT OpenCourseWare)
- Licence
- CC BY-NC-SA 4.0, as MIT OpenCourseWare states it. Non-commercial use only.
- Attribution
- Prof. Arthur Mattuck, Prof. Haynes Miller, Dr. Jeremy Orloff, Dr. John Lewis. Differential Equations. Fall 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu/. License: Creative Commons BY-NC-SA.
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- Tweens profiles and older. How courses are placed
All the lectures
- Introduction to the 18.03 Applets
- Separable Equations
- Geometric View of DE's: Direction Fields, Integral Curves
- Example, Existence and Uniqueness
- Example Where Existence and Uniqueness Fails
- Exploration of the Isoclines Applet
- Direction Fields
- Euler's Method
- Example of Euler's Method
- Exploration of the Euler's Method Applet
- Better Methods
- Pitfalls
- Euler's Method
- Linear First Order ODE's: Definition
- Linear First Order ODE's: Examples
- Linear Equations
- Solutions to Linear First Order ODE's
- Solutions of First Order Linear Equations
- Complex Arithmetic, Euler's Formula
- Justification of Euler's Formula
- Polar Form
- Example: Complex Replacement
- Nth Roots
- Complex Numbers and Euler's Formula
- Three Proofs of the Basic Trigonometric Identity
- Sinusoidal Functions
- Solutions to Constant Coefficient Linear First Order ODE's
- Examples of Constant Coefficient Linear First Order ODE's
- More Examples of Constant Coefficient Equations
- First Order Constant Coefficient Linear ODE's
- Exploration of the Amplitude and Phase: First Order Applet
- Sinusoidal Inputs
- First Order Autonomous Equations
- Bank Account
- Example: Logistic Model
- Exploration of the Phase Lines Applet
- Example: Logistic Model with Harvesting
- Autonomous Equations and Phase Lines
- Constant Coefficient Second Order Homogeneous DE's
- Real Roots: Overdamping
- Complex Roots: Underdamping
- Repeated Roots: Critical Damping
- Real Solutions from Complex Roots
- Homogeneous Constant Coefficient Equations: Real Roots
- Homogeneous Constant Coefficient Equations: Any Roots
- Damping and Pseudo-Frequency
- Exploration of the Damped Vibrations Applet
- Damped Harmonic Oscillators
- Linear Operators and Superposition
- Inhomogeneous Second Order DE's: Definitions and Models
- Particular and General Solutions
- Inhomogeneous DE's with Exponential Input
- Forced Oscillations
- Definition of Stability
- Stability of Constant Coefficient Second Order DE's
- Gain and Phase Lag
- Undetermined Coefficients
- Solving DE's with Exponential Input when p(a) = 0
- Resonance in the Undamped System
- Reconciling the Resonant and Non-resonant Solutions
- Pure Resonance
- Exploration of the Amplitude and Phase: Second Order II Applet
- Frequency Response
- Introduction to Fourier Transform
- Fourier Series for Functions with Period 2L
- Orthogonality Relations
- Computing Fourier Series
- Even and Odd Functions
- Convergence of Fourier Series
- Exploration of the Fourier Coefficient Applet
- Manipulating Fourier Series
- Solving ODE's, Resonant Frequencies
- Hearing I
- Hearing II
- Linear ODE's with Periodic Input
- Step and Delta Functions: Integration and Generalized Derivatives
- Unit Step and Impulse Response
- Example: f(t)*1
- Example: Radioactive Dumping
- Exploration of the Convolution Accumulation Applet
- Convolution and Green's Formula
- Laplace Transform
- Notation and Linearity
- Start of Laplace Table
- Functions of Exponential Type
- Laplace Transform of Powers
- Laplace Transform: Basics
- Heaviside Coverup Method
- Partial Fractions and Laplace Inverse
- Laplace: Solving ODE's
- Laplace Transform of Convolution
- Pole Diagrams
- Introduction to Systems
- Autonomous Systems: Solutions are Parametrized Curves
- Linear Systems of Equations
- Matrix Methods Introduction, First Example
- Matrix Systems: General Terminology
- Complex Eigenvalues: Example
- Three Chamber Mixing Example: Repeated Roots
- Linear Systems: Matrix Methods
- Linear Systems: Complex Roots
- Sketching Solutions
- More Sketching Solutions
- Trace-Determinant Diagram
- Phase Portraits
- Theory of Systems
- Fundamental Matrices
- The Matrix Exponential I
- The Matrix Exponential II
- Example: Solving a System with the Matrix Exponential
- Calculating the Exponential Matrix
- Variation of Parameters
- Matrix Exponentials
- Decoupling
- The Relation Between Nonlinear Systems and First-order ODE's
- The Nonlinear Pendulum
- Example: Predator-Prey and Volterra's Principle
- Linearization
- Limit Cycles
- Existence of Limit Cycles
- Non-existence of Limit Cycles
- A Story about Limit Cycles