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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Growth and Decay | youtube icon Topic Playlist

One of the simplest models for a differential equation is when the rate of growth or decay of a thing is proportional to the current quantity of the thing. The mass of a radioactive substance over time and the population of an organism over time are typical examples.

Radioactive Decay | youtube icon Explanation Video

The rate of decay of a radioactive substance is proportional to the mass of the radioactive substance. The typical measurement is the half-life, which is the length of time that half of the radioactive substance will decay. The ODE is typically written as,

\begin{equation} Q' = -kQ \end{equation}

where $Q(t)$ is the mass of radioactive substance at time $t$ and $k$ is the rate of radioactive decay. The convention is for $k$ to be a positive number, so the RHS is written as $-kQ$ meaning the rate of change, $Q'$, is always negative.

Radioactive Decay Example 1 | youtube icon Solution Video

Draw a direction field for the radioactive decay of a substance assuming you start with $Q(0) > 0$ for a positive mass and decay rate $k=0.5$.

\begin{equation} Q' = -0.5Q \end{equation}

Radioactive Decay Example 2 | youtube icon Solution Video

Solve the initial value problem.

\begin{equation} Q' = -0.5Q \qquad Q(0) = 10 \end{equation}

Radioactive Decay Example 3 | youtube icon Solution Video

Determine the half-life for the following ODE for the radioactive decay of a substance.

\begin{equation} Q' = -3Q \qquad Q(0) = Q_{0} \end{equation}

Population Growth | youtube icon Explanation Video

The rate of growth of a organism, such as bacteria in milk, can be modelled as proportional to the current amount of bacteria in the milk. The model works well as long as the amount of bacteria is small compared the the quantity of milk. The ODE is typically written as,

\begin{equation} P' = kP \end{equation}

where $P(t)$ is the mass of radioactive substance at time $t$ and $k$ is the rate of population growth. The convention is for $k$ to be a positive number, so the RHS $P'$ is always positive because the population of bacteria in the milk is always increasing.

Population Growth Example 1 | youtube icon Solution Video

Draw a direction field for the population growth of bacteria with initial population $P(0) > 0$ and growth rate $k=10$.

\begin{equation} P' = 10P \end{equation}

Population Growth Example 2 | youtube icon Solution Video

Solve the initial value problem.

\begin{equation} P' = 10P \qquad P(0) = 0.7 \end{equation}

Population Growth Example 3 | youtube icon Solution Video

Solve for $k$ if $P(0)=3$ and $P(2)=8$.

\begin{equation} P' = kP \qquad P(0)=3 \quad P(2)=8 \end{equation}