Interactive Course

Differential Equations II

The frontier of scientific modeling.

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Overview

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.

Topics covered

  • Bessel's equation
  • Chaos and the Lorenz attractor
  • Conformal maps
  • Equilibria and limit cycles
  • Fourier and Laplace transforms
  • Nonlinear systems
  • Partial differential equations
  • Power series solutions
  • Predator-prey models
  • Schrödinger's equation and the hydrogen atom
  • Separation of variables
  • Wave and heat equations

Interactive quizzes

36

Concepts and exercises

440+

Course map

Prerequisites and Next Steps

  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?

      1
    2. PDEs in a Nutshell

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

      2
  2. 2

    Nonlinear Equations

    The essential toolbox for systems of nonlinear equations.

    1. Lotka-Volterra I

      Learn how to visualize nonlinear pairs with a direction field.

      3
    2. Lotka-Volterra II

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

      4
    3. Linearization

      Use linear algebra to figure out the nature of equilibria.

      5
    4. The Hartman-Grobman Theorem

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

      6
    5. Application: Get Pumped for Lasers!

      Sharpen your nonlinear skills on a space age application.

      7
    6. Challenge: Liapunov Functions

      Learn how to classify nonlinear equilibria with energy-like functions.

      8
    7. Cycles and The Pursuit of Happiness

      Find out what limit cycles are all about with a classic pursuit problem.

      9
    8. The Poincaré-Bendixson Theorem

      Deduce the existence of a limit cycle in a real-world chemical oscillator.

      10
    9. Chaos and the Lorenz Equations

      Discover how nonlinear equations changed our views on science and predictability forever.

      11
  3. 3

    Partial Differential Equations

    The basics of differential equations with many variables.

    1. 1D Waves & d'Alembert's Formula

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

      12
    2. Sources & Boundary Conditions

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

      13
    3. Challenge: 2D & 3D Waves

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

      14
    4. Separation of Variables & Waves

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

      15
    5. Fundamental Solutions

      Investigate and master the equation which governs diffusion and heat conduction.

      16
    6. Challenge: Fun with Functionals

      Discover how to squeeze useful info out of a pde without actually solving it.

      17
    7. Laplace's Equation

      Reconstruct a pde common in science and engineering from a classic geometry problem.

      18
    8. Approximating Laplace

      How can computers help solve a pde?

      19
  4. 4

    Transform Methods

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

    1. The Fourier Transform

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

      20
    2. Practice: Fourier & The Heat Equation

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

      21
    3. Practice: Fourier & Laplace's Equation

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

      22
    4. Challenge: Fourier & 3D Waves

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

      23
    5. Schrödinger's Equation

      Find deeper meaning in the Fourier transform by going on a trip to the quantum realm.

      24
    6. Conformal Maps

      Sometimes, it's not the equation that needs to change, it's space itself!

      25
    7. The Laplace Transform

      Come full circle and see how transform methods work on ordinary differential equations.

      26
    8. Laplace Transform Applications

      Apply the Laplace transform to interesting electrical engineering problems.

      27
  5. 5

    Power Series

    Discover the power of unending sums.

    1. Series Solutions I

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

      28
    2. Series Solutions II

      Explore some quirks and special features of infinite series.

      29
    3. The Airy Equation

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

      30
    4. Interlude: Return of the Wronskian

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

      31
    5. Cauchy-Euler Equation

      Practice power series method by working on a fluid dynamics problem.

      32
    6. Bessel's Equation

      Apply the experience you gained solving the Cauchy-Euler equation to model a circular vibrating drumhead.

      33
    7. Hermite's Equation

      Return to the subatomic realm and use power series to solve the quantum oscillator.

      34
    8. Capstone: Hydrogen Atom I

      Pull everything together to reproduce one of the greatest scientific achievements of the 20th century.

      35
    9. Capstone: Hydrogen Atom II

      Complete the hydrogen atom model and sketch an electron orbital, a staple of chem classes the world over.

      36