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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Repeated Real Root | youtube icon Topic Playlist

Sometimes the solutions to differential equations were guessed by mathematicians. There is no derivation. For the ODE $ay''+by'+cy=0$ someone guessed a solution $y(x)=e^{rx}$ then tried to figure out what value $r$ needed to be to solve the differential equation. The result of plugging in $e^{rx}$ for $y(x)$ is $e^{rx}(ar^{2}+br+c)=0$. The polynomial $ar^{2}+br+c$ is called the characteristic polynomial and solving for the roots gives the values of $r$ to solve the ODE.

Characteristic polynomial when there is a repeated real root | youtube icon Explanation Video

One of three cases for the characteristic polynomial is a repeated real root.

\begin{equation} ay''+by'+cy=0 \qquad y(x)=e^{rx} \end{equation} \begin{equation} e^{rx}(ar^{2}+br+c)=0 \qquad ae^{rx}(r-r_{1})(r-r_{2})=0 \end{equation}

Where $r_{1}$ and $r_{2}$ are the roots of the characteristic polynomial. When $r_{1} = r_{2}$, the solutions to the ODE are,

\begin{equation} y_{1}(x)=e^{r_{1}x}=e^{r_{2}x} \qquad y_{2}(x)=xe^{r_{1}x}=xe^{r_{2}x} \end{equation}

The general solution is,

\begin{equation} y(x)=c_{1}e^{r_{1}x}+c_{2}xe^{r_{1}x}=c_{1}e^{r_{2}x}+c_{2}xe^{r_{2}x} \end{equation}

Example 1 | youtube icon Solution Video

Find the general solution to the ODE.

\begin{equation} y'' - 8y' + 16y = 0 \end{equation}

Example 2 | youtube icon Solution Video

Solve the initial value problem.

\begin{equation} y'' - 2y' + y = 0 \qquad y(0) = 2 \qquad y'(0)=-1 \end{equation}

Example 3 | youtube icon Solution Video

Solve the IVP for a mass spring damper system.

\begin{equation} z'' + 6z' + 9z = 0 \qquad z(0) = 3 \qquad z'(0) = -2 \end{equation}