Home / Courses / Differential Equations

Differential Equations

Free Differential Equations study guide: 10 units with summaries, key ideas, flashcards and 182 practice questions with step-by-step solutions.

Lines up with: AP Calculus BC Unit 7 (intro only); GRE Mathematics Subject Test.

Units

1. UNIT 01

   First-order equations

   First-order differential equations involve only y and y′. Most you'll see are separable, linear or exact, and each type has its own method.
2. UNIT 02

   Modeling with first-order DEs

   Differential equations model change in the real world. Set up the rate of change in words first, then translate.
3. UNIT 03

   Numerical methods

   When you can't solve an equation exactly, numerical methods approximate the solution step by step.
4. UNIT 04

   Second-order linear equations

   Second-order linear equations with constant coefficients are solved with the characteristic equation. The type of roots decides the form of the solution.
5. UNIT 05

   Nonhomogeneous equations

   For nonhomogeneous equations, the general solution is the homogeneous solution plus one particular solution.
6. UNIT 06

   Mechanical vibrations

   Springs and circuits follow the same second-order equation. Damping and forcing decide whether the motion dies out, oscillates or blows up.
7. UNIT 07

   Laplace transforms

   The Laplace transform turns a differential equation into an algebra problem. Solve for the transform, then invert.
8. UNIT 08

   Systems of first-order DEs

   Systems of first-order equations model several interacting quantities. Linear systems are solved with eigenvalues and eigenvectors.
9. UNIT 09

   Series solutions (optional)

   Power-series solutions handle equations with variable coefficients where other methods fail.
10. UNIT 10

    Fourier series and partial differential equations

    Fourier series write periodic functions as sums of sines and cosines, which lets us solve the heat and wave equations by separation of variables.