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Differential Equations I

The math of change, from economics to physics.

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.

Interactive
quizzes

30

Concepts and
exercises

400+
  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?

    2. Our First Equation

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

  2. 2

    First-Order Separable Equations

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

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      Modeling: The Drag Equation

      Make a differential equation through experimentation.

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      Separate and Integrate

      Conquer first-order separable equations with a single technique.

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      Application: Molecular Motor

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

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      The Phase Portrait

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

  3. 3

    Advanced First-Order Equations

    Learn the power of combining geometry and differential equations

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      Direction Fields

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

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      Integrating Factor

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

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      Application: A Mixing Problem

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

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      The Potential

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

  4. 4

    Basics of Linear Systems

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

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      The Phase Plane

      Learn to visualize the essential features of equation pairs.

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      The Matrix Exponential

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

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      Application: Underdamped Springs

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

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      Non-Diagonalizable Matrices

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

  5. 5

    Higher-Order Equations

    The workhorse of physics, engineering, and applied mathematics.

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      Equations of Order Two

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

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      Application: RLC Filter

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

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      Challenge: Higher-Order Equations

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

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      Application: Hangin' Around

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