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

Direction Fields | Topic Playlist

Given an ODE $y'=f(x,y)$, a direction field is a graph in the $xy$-plane with individual lines drawn at many points. The points are usually drawn at the vertices of a grid. The slope of a line drawn at point $(x,y)$ has the slope determined by the ODE at point $(x,y)$.

Example 1 | Solution Video

Example 2 | Solution Video

Example 3 | Solution Video