Differential Equations II

The frontier of scientific modeling.

Drum vibrations, heat flow, the quantum nature of matter, and the dynamics of competing species are just a few real-world examples involving advanced differential equations.

These models and many others from across the sciences, engineering, and finance have nonlinear terms or several independent variables. Their equations hold many surprises, and their solutions draw on other areas of math like linear algebra and vector calculus.

Interactive
quizzes

36

Concepts and
exercises

440+
  1. 1

    Introduction

    What are nonlinear and partial differential equations?

    1. Nonlinear Equations in a Nutshell

      What makes an equation nonlinear, and why does it matter?

    2. PDEs in a Nutshell

      Tour the equations governing heat, waves, diffusion, and the quantum realm.

  2. 2

    Nonlinear Equations

    The essential toolbox for systems of nonlinear equations.

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      Lotka-Volterra I

      Learn how to visualize nonlinear pairs with a direction field.

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      Lotka-Volterra II

      Apply multivariable calculus ideas to an important pair of nonlinear equations.

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      Linearization

      Use linear algebra to figure out the nature of equilibria.

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      The Hartman-Grobman Theorem

      Find out why linearization works so well by borrowing ideas from topology.

  3. 3

    Partial Differential Equations

    The basics of differential equations with many variables.

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      1D Waves & d'Alembert's Formula

      Begin your journey into partial differential equations (pdes) with waves on a length of rope.

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      Sources & Boundary Conditions

      What happens when a 1D wave reaches the end of its rope?

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      Challenge: 2D & 3D Waves

      Step into higher dimensions by modeling surface ripples and sound waves.

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      Separation of Variables & Waves

      Learn how to split a difficult linear pde into a set of easier problems.

  4. 4

    Transform Methods

    Learn how to turn hard diff eq problems into easy ones.

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      The Fourier Transform

      Find out what signals analysis and linear pdes have in common.

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      Practice: Fourier & The Heat Equation

      Learn to wield the power of the Fourier transform by practicing on the heat equation.

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      Practice: Fourier & Laplace's Equation

      Continue to develop your Fourier transform skills with Laplace's equation.

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      Challenge: Fourier & 3D Waves

      Rise to the challenge of modeling 3D compression waves with the Fourier transform.

  5. 5

    Power Series

    Discover the power of unending sums.

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      Series Solutions I

      Learn how infinite sums clear the path to solving difficult differential equations.

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      Series Solutions II

      Explore some quirks and special features of infinite series.

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      The Airy Equation

      Use power series to solve an important problem from quantum mechanics.

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      Interlude: Return of the Wronskian

      What's a reliable way for telling power series apart?