Interactive Course

Differential Equations I

The math of change, from economics to physics.

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Overview

Differential equations show up in just about every branch of science, including classical mechanics, electromagnetism, circuit design, chemistry, biology, economics, and medicine.

From analyzing the simple harmonic motion of a spring to looking at the population growth of a species, differential equations come in a rich variety of different flavors and complexities.

Topics covered

  • Applications in Engineering
  • Applications in Physics
  • Direction Fields
  • Euler's Method
  • Integrating Factors
  • Linear Systems
  • Matrix Exponential
  • Modeling
  • Perturbation Method
  • Phase Portraits
  • Separable First-Order Equations
  • Wronskian Determinants

Interactive quizzes

31

Concepts and exercises

390+

Course map

Prerequisites and Next Steps

  1. 1

    Introduction

    Discover why differential equations are so important!

    1. Differential Equations in a Nutshell

      What are differential equations and what are they used for?

      1
    2. Our First Equation

      Use Newton's Law of Cooling to build and solve your first differential equation.

      2
    3. A Quick Review

      Review the algebra and calculus tools you'll need for the road ahead.

      3
  2. 2

    First-Order Separable Equations

    Build and explore real-world models with first-order equations.

    1. Modeling: The Drag Equation

      Make a differential equation through experimentation.

      4
    2. Separate and Integrate

      Conquer first-order separable equations with a single technique.

      5
    3. Application: Molecular Motor

      Unravel secrets of the biological micro-world with first-order equations.

      6
    4. The Phase Portrait

      Learn to sketch integral curves like a pro by solving this chemistry challenge.

      7
    5. Concavity and Partial Fractions

      Improve your solution curves and add an important integration skill to your toolbox.

      8
    6. Application: In the Chem Lab

      Practice your first-order skills while exploring this chemistry application.

      9
    7. Capstone: Vampires of Cancelvania

      Use differential equations to save the countryside from a vampire plague!

      10
  3. 3

    Advanced First-Order Equations

    Learn the power of combining geometry and differential equations

    1. Direction Fields

      Use a simple, geometric technique to draw integral curves on the phase plane.

      11
    2. Integrating Factor

      Combine vectors and geometry into a technique for solving first-order problems.

      12
    3. Application: A Mixing Problem

      Apply the integrating factor to a classic real-world problem.

      13
    4. The Potential

      Explore the meaning of the integrating factor in the context of physics.

      14
    5. Application: Fluid Flow

      Explore the potential for using first-order equations to make real-world models.

      15
    6. Capstone: The Great Escape

      Drop in on a marooned astronaut who uses first-order equations to call for rescue.

      16
  4. 4

    Basics of Linear Systems

    Matrices and vectors unlock the gateway to higher-order equations.

    1. The Phase Plane

      Learn to visualize the essential features of equation pairs.

      17
    2. The Matrix Exponential

      Learn how to use vector geometry as a powerful tool for exploring linear systems.

      18
    3. Application: Underdamped Springs

      Gain hands-on experience with the matrix exponential in a familiar setting.

      19
    4. Non-Diagonalizable Matrices

      What do you do when you just don't have enough eigenvectors?

      20
    5. Review: Math of a Salesman

      Practice your linear systems skills on a mind-altering robot sociology problem.

      21
    6. Nonhomogeneous Systems: Part I

      Make an atom smasher by combining physics with systems of equations.

      22
    7. Nonhomogeneous Systems: Part II

      Make a master formula for linear systems and complete your cyclotron.

      23
    8. Challenge: Floquet Theory

      Strengthen your grasp on linear systems while learning how medical NMRI works.

      24
  5. 5

    Higher-Order Equations

    The workhorse of physics, engineering, and applied mathematics.

    1. Equations of Order Two

      Uncover the deep connection between higher-order equations and systems.

      25
    2. Application: RLC Filter

      Find out what second-order equations can do for electrical engineers.

      26
    3. Challenge: Higher-Order Equations

      Reach beyond second-order and uncover a master formula with the Wronskian matrix.

      27
    4. Application: Hangin' Around

      Solve a structural engineering problem with a third-order equation.

      28
    5. Application: Beam Me Up!

      Can a building withstand an earthquake? Find out with fourth-order equations!

      29
    6. Application: Get Your Motor Runnin'

      Retrieve a treasure from the briny depths with a third-order equation.

      30
    7. Challenge: Why's The Sky Blue?

      Answer a very common science question by applying the perturbation method.

      31