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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Variation of Parameters | youtube icon Topic Playlist

Variation of Parameters is a method to find the particular solution of a non-homogeneous linear ODE when the homogeneous solutions are known, which is more powerful than the Method of Undetermined Coefficients since this can solve for any non-homogeneous term if the integrals can be evaluated. The formula requires the homogenous solutions and evaluating some integrals. This page is focused on second order ODEs, but the method applies to higher order ODEs as well.

Variation of Parameters Formula | youtube icon Explanation Video

Given a non-homogeneous linear ODE with its two homogeneous solutions $y_{1}(x)$ and $y_{2}(x)$,

\begin{equation} y''+p(x)y'+q(x)y=g(x) \end{equation}

The formula for the particular solution $y_{p}(x)$ is,

\begin{equation} y_{p}(x) = - y_{1}(x)\int \frac{y_{2}(x)g(x)}{W(y_{1},y_{2})} \text{ d}x + y_{2}(x)\int \frac{y_{1}(x)g(x)}{W(y_{1},y_{2})} \text{ d}x \end{equation}

Where $W(y_{1},y_{2})$ is the Wrońskian.

Example 1 | youtube icon Solution Video

Find the particular solution to the following differential equation.

\begin{equation} x^{2}y''-2xy'+2y=\sqrt{x} \qquad y_{1} = x \quad y_{2} = x^{2} \end{equation}

Example 2 | youtube icon Solution Video

Find the particular solution to the following differential equation.

\begin{equation} y'' + 4y = \tan(x) \qquad y_{1} = \cos(2x) \quad y_{2} = \sin(2x) \end{equation}

Example 3 | youtube icon Solution Video

Find the particular solution to the following differential equation.

\begin{equation} x^{2}y'' - 2xy' - (x^{2}-2)y = 3x^{3} \qquad y_{1} = e^{x} \quad y_{2} = e^{-x} \end{equation}

Variation of Parameters Derivation | youtube icon Explanation Video

Given a non-homogeneous linear ODE with its two homogeneous solutions $y_{1}(x)$ and $y_{2}(x)$,

\begin{equation} y''+p(x)y'+q(x)y=g(x) \end{equation}

The derivation of the formula is based on two assumptions that just happen to work. The first assumption is to make up two unknown functions $u_{1}(x)$ and $u_{2}(x)$ that satisfy the following equation.

\begin{equation} y_{p}(x) = u_{1}(x)y_{1}(x) + u_{2}(x)y_{2}(x) = u_{1}y_{1} + u_{2}y_{2} \end{equation}

Since there's one equation but two unknown functions $u_{1}(x)$ and $u_{2}(x)$, another equation is assumed so that there are two equations and two unknowns, so there could still be a way to solve for the unknown $u_{1}(x)$ and $u_{2}(x)$ functions.

\begin{equation} u_{1}'y_{1} + u_{2}'y_{2} = 0 \end{equation}