Table of Contents
Differential Equations
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Courses

  1. Linear Algebra
  2. Multivariable Calculus
  3. Differential Equations
  4. Miscellaneous Topics

Differential Equations

  1. Introduction
    1. Notation and Definitions
    2. Verifying Solutions
    3. Initial Values Problems
  2. First Order Differential Equations
    1. Direction Fields
    2. Equilibria, Stability, and Phase Lines
    3. Separable Equations
    4. Integrating Factor
    5. Bernoulli Equations and Substitutions
    6. Exact Equations
    7. Exact Equations with Integrating Factor
    8. Euler's Method
    9. Existence and Uniqueness
    10. Interval of Validity
    11. Applications
      1. Radioactive Decay and Population Growth
      2. Mixing Tank Problem
      3. Terminal Velocity
      4. Continuous Compound Interest
  3. Wrońskian
  4. Second Order Differential Equations
    1. $ay''+by'+cy = 0$
      1. Distinct Real Roots
      2. Repeated Real Root
      3. Imaginary Roots
    2. Spring-Mass-Damper
      1. Equivalent RLC Circuit
      2. Underdamped, Overdamped, Critically Damped
      3. Undamped Oscillations and Resonance
    3. Method of Undetermined Coefficients
    4. Reduction of Order
    5. Variation of Parameters
    6. Cauchy-Euler Equations
  5. Laplace Transform
    1. Lookup Table and WolframAlpha
  6. Systems of ODEs
    1. Simplest Case $\begin{bmatrix}x' \\ y'\end{bmatrix} = \begin{bmatrix}a & b \\ c & d \end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}$
      1. Poincaré Diagram
  7. Preview of Dynamical Systems
    1. Self-Study
    2. Rössler Attractor

Additional Resources

  1. WolframAlpha Examples
  2. Direction Field Plotter by Ariel Barton
  3. Phase Plane Plotter by Ariel Barton


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Underdamped, Overdamped, and Critically Damped Spring Mass Damper System | youtube icon Topic Playlist

A mass-spring-damper with no forcing term has three solution behaviours called underdamped, overdamped, and critically damped. An underdamped system oscillates about the equilibrium and is slow to decay to equilibrium. An overdamped system decays to the equilibrium without oscillating. A critically damped system separates the underdamped and overdamped cases, and solutions move as quickly as possible toward equilibrium without oscillating about the equilibrium.

Underdamped

This is the imaginary root case. A practical application is airplane wings that oscillate can be made lighter versus a stiffer construction.

underdamped plots

Overdamped

This is the two distinct real roots case. This is desirable in systems where overshooting could be disastrous, such as extremely large and heavy doors with built in dampers.

overdamped plots

Critically Damped

This is the case separating underdamped and overdamped at a specific damping coefficient relative to the mass and spring constant. This can be desirable in automobile suspensions driving over a bump so there are no oscillations but with minimum stiffness.

critically damped plots