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 [x′y′]=[abcd][xy]
      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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RLC Circuit | youtube icon Topic Playlist

The ideal models of a resistor, capacitor, and inductor (RLC) connected in series with a battery result in an ODE with the same form as a spring-mass-damper system. All the behaviours and analysis can be applied directly to analysing an RLC circuit in series.

Circuit Diagram and ODE | youtube icon Explanation Video

For the following RLC circuit,

RLC circuit diagram

The ODE governing its behaviour is,

LQ″+RQ′+1CQ=E(t)

where R, L, C, and E(t) are the resistance, inductance, capacitance, and impressed voltage as a function of time.

Most math textbooks do not explain the physics of the model in depth and instead just state equations to use for each component. There are many physics based explanations online if that is of interest.

ODE Derivation | youtube icon Solution Video

Make the ODE for the RLC circuit given the following relationships:

Voltage drop across a capacitor =Q(t)C where Q(t) is the charge stored on the capacitor.

Voltage drop across a resistor =RI(t) where I(t) is the current flowing through the resistor.

Voltage drop across an inductor =LdIdt=LI′(t).

The rate of change of charge, Q′(t), on the capacitor is the current I(t).

Kirchhoff's Voltage Law states the sum of voltages around a loop is zero.