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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Mixing Tank Problems | youtube icon Topic Playlist

The most common differential equation mixing problem is a tank of water with salt water being poured in at the same rate water is flowing out of the tank. There are a few variations which are all typical homework exercises in an introductory differential equations course. A real world application is polluted water flowing into a large lake which also has an outlet, solving the ODE to determine how polluted the water will be over time.

Example 1 | youtube icon Solution Video

A tank initially containing 500L of fresh water has salt water flowing into it at a rate of 10L per minute with a concentration of 0.3g of salt per 1L of water. The tank has a drain at the bottom also flowing at 10L per minute. Write an ODE for the g of salt $Q(t)$ in the tank at time $t$ where $t$ is in minutes. Assume the tank is thoroughly mixed so the salt in the tank is even distributed throughout the entire tank.

Example 2 | youtube icon Solution Video

A tank initially containing 500L of fresh water has salt water flowing into it at a rate of 10L per minute with a concentration of 0.3g of salt per 1L of water. The tank has a drain at the bottom also flowing at 10L per minute. When will the concentration of salt in the tank be within 2% of the limiting value?

Example 3 | youtube icon Solution Video

A tank initially containing 700L of fresh water has salt water flowing into it at a rate of 8L per minute with the concentration of salt varying over time given by $2 + \sin(t)$ g salt per 1L of water. The tank has a drain at the bottom also flowing at 8L per minute. Write an ODE for the g of salt $Q(t)$ in the tank at time $t$.

Example 4 | youtube icon Solution Video

A tank initially containing 300L of fresh water has salt water flowing into it at a rate of 5L per minute with a concentration of 0.4g of salt per 1L of water. The tank has a drain at the bottom also flowing at 5L per minute. After 10 minutes, the salt water flowing into the tank is replaced with fresh water. Calculate how much salt is in the tank 5 minutes after fresh water started flowing into the tank.

Example 5 | youtube icon Solution Video

A tank initially containing 250L of fresh water has salt water flowing into it at a rate of 3L per minute with a concentration of 0.25g of salt per 1L of water. The tank has a drain at the bottom flowing at 1L per minute, so the volume of water in the tank is increasing over time. Write an ODE for the g of salt $Q(t)$ in the tank at time $t$.