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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Terminal Velocity | youtube icon Topic Playlist

The force of drag on a falling object can be approximated as proportional to the current velocity of the object. This can be written as a first order ordinary differential equation. The terminal velocity is when the force of drag is equal and opposite to the force of gravity and the velocity of the falling object does not change.

Sum of Forces on Falling Object | youtube icon Explanation Video

The force of gravity and the force of drag sum to equal mass times acceleration from Newton's Law with mass $m$ and drag coefficient $\gamma$, where $v(t)$ is the velocity at time $t$ and the drag on the mass is proportional to its velocity.

\begin{equation} mv' = -mg - \gamma v \end{equation}

In this ODE, positive velocity means the object is gaining altitude and negative velocity means the object is falling towards the ground.

Example 1 | youtube icon Solution Video

Write the ODE for a mass $m = 4\text{kg}$, and drag coefficient $k = 0.5\dfrac{\text{kg}}{\text{s}}$. Draw a direction field and determine the terminal velocity. Use $g = 9.8\dfrac{\text{m}}{\text{s}^{2}}$ for Earth's gravity and assume drag is proportional to velocity.

Example 2 | youtube icon Solution Video

Solve the following ODE for a falling object and calculate the limit as $t$ goes to infinity to determine the terminal velocity.

\begin{equation} 4v' = -39.2 - 0.5 v \end{equation}