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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Euler's Method | youtube icon Topic Playlist

Steps to Applying Euler's Method | youtube icon Explanation Video

Given an IVP with a step size

\begin{equation} y' = f(x,y) \qquad y(x_{0}) = y_{0} \qquad \text{step size } \Delta x \end{equation}

The next $x_{n+1} = x_{n} + \Delta x$ and the next $y_{n+1} = y_{n} + \Delta x f(x_{n},y_{n})$ and you iterate.

\begin{align} y_{new} &= y_{old} + (\text{change in $x$}) \frac{\text{change in $y$}}{\text{change in $x$}} \\ &= y_{old} + (\text{change in $y$}) \end{align}

Example 1 | youtube icon Solution Video

Apply Euler's method to the ODE.

\begin{equation} \frac{\text{d}y}{\text{d}t} = \frac{1}{2}y \qquad y(0) = 1 \qquad \Delta t = 1 \end{equation}

Check your work using WolframAlpha.

Example 2 | youtube icon Solution Video

Apply Euler's method to the ODE.

\begin{equation} \frac{\text{d}y}{\text{d}x} = -xy \qquad y(0) = 1 \qquad \Delta x = 0.2 \end{equation}

Check your work using WolframAlpha.

Example 3 | youtube icon Solution Video

Apply Euler's method to the ODE.

\begin{equation} \frac{\text{d}y}{\text{d}t} = y(3-y) \qquad y(2) = 1 \qquad \Delta t = 0.1 \end{equation}

Check your work using WolframAlpha.