Advanced Math · Level 5

5.2 Differential Equations

Go on a grand tour of differential equations and get the hands-on experience needed to master the essentials.

Differential Equations in a Nutshell

Our First Equation

Modeling: The Drag Equation

Separate and Integrate

Application: Molecular Motor

The Phase Portrait

Concavity and Partial Fractions

Application: In the Chem Lab

Capstone: Vampires of Cancelvania

Direction Fields

Integrating Factor

Application: A Mixing Problem

The Potential

Application: Fluid Flow

Capstone: The Great Escape

The Phase Plane

The Matrix Exponential

Application: Underdamped Springs

Non-Diagonalizable Matrices

Review: Math of a Salesman

Nonhomogeneous Systems: Part I

Nonhomogeneous Systems: Part II

Challenge: Floquet Theory

Equations of Order Two

Application: RLC Filter

Challenge: Higher-Order Equations

Application: Hangin' Around

Application: Beam Me Up!

Application: Get Your Motor Runnin'

Challenge: Why's The Sky Blue?

Course description

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. This course takes you on a grand tour of some of the most important differential equations of the natural sciences, giving you the hands-on experience needed to master the essentials.

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

Prerequisites and next steps

You’ll need a grasp of the basic derivative and integration rules covered in a first semester calculus course. Integration techniques are useful to know, but not necessary. Knowing basic linear algebra up to eigenvalues and eigenvectors and multivariable calculus up to gradients is essential.


  • Multivariable Calculus
  • Introduction to Linear Algebra