Differential Equations | Index

A differential equation links a function to its own rates of change, so differential equations describe how things move, grow, cool and vibrate.

The course covers first-order equations, second-order linear equations, Laplace transforms, systems of equations, series solutions, and an introduction to Fourier series and partial differential equations.

It’s for college students in a first course in ordinary differential equations, usually taken after Calculus 2 or Calculus 3.

What to know first: Calculus 2, especially integration techniques and power series, plus partial derivatives from Calculus 3, Unit 3. Separable equations and slope fields are in Calculus 2, Unit 4. The unit on systems uses matrices from Precalculus, Unit 4 and eigenvalues from Linear Algebra (coming soon).

Lessons for this course are coming soon. The list below shows every planned lesson in teaching order.

Units in this course

Unit 1: First-Order Differential Equations

  1. Classifying Differential Equations (coming soon) — order, linear or nonlinear, and ordinary or partial
  2. First-Order Linear Differential Equations (coming soon) — solve linear equations with an integrating factor
  3. Exact Differential Equations (coming soon) — test for exactness and find the potential function
  4. Bernoulli Equations (coming soon) — turn a Bernoulli equation into a linear equation with a substitution
  5. Substitution for Homogeneous Equations (coming soon) — solve first-order equations of the form dy/dx = F(y/x) with the substitution y = vx
  6. Existence and Uniqueness Theorem (coming soon) — know when an initial-value problem has exactly one solution
  7. Autonomous Equations and Phase Lines (coming soon) — find equilibrium solutions and decide whether they are stable
  8. Mixing Problems (coming soon) — set up and solve tank mixing problems with first-order equations
  9. Runge-Kutta Method (coming soon) — get more accurate numerical solutions than Euler’s method gives

Also see: slope fields, Euler’s method and separable equations in Calculus 2, Unit 4.

Unit 2: Second-Order Linear Equations

  1. Second-Order Homogeneous Linear Equations (coming soon) — solve with the characteristic equation when its roots are real and different
  2. The Wronskian (coming soon) — check whether two solutions form a fundamental set
  3. Reduction of Order (coming soon) — find a second solution from one known solution
  4. Complex Roots of the Characteristic Equation (coming soon) — write real solutions with sines and cosines
  5. Repeated Roots of the Characteristic Equation (coming soon) — find a second solution when the root repeats
  6. Method of Undetermined Coefficients (coming soon) — find a particular solution by guessing its form
  7. Variation of Parameters (coming soon) — find a particular solution when undetermined coefficients doesn’t apply
  8. Cauchy-Euler Equations (coming soon) — solve equations like ax²y″ + bxy′ + cy = 0, where each power of x matches the order of its derivative
  9. Spring-Mass Systems (coming soon) — model free vibrations, with and without damping
  10. Forced Oscillations and Resonance (coming soon) — model a spring or circuit pushed by an outside force

Unit 3: Laplace Transforms

  1. Laplace Transform (coming soon) — the definition and a table of common transforms
  2. Inverse Laplace Transform (coming soon) — find the original function with tables and partial fractions
  3. Solving Differential Equations with Laplace Transforms (coming soon) — solve initial-value problems by working with transforms
  4. Unit Step Functions (coming soon) — model switches and piecewise forcing with the Heaviside function
  5. Dirac Delta Function (coming soon) — model a sudden impulse, like a hammer strike
  6. Convolution Theorem (coming soon) — find the inverse transform of a product of transforms

Also see: partial fraction decomposition in Precalculus, Unit 2, and improper integrals in Calculus 2, Unit 2.

Unit 4: Systems of Differential Equations

  1. Systems of Differential Equations (coming soon) — write models and higher-order equations as first-order systems
  2. Solving Linear Systems with Eigenvalues (coming soon) — solve x′ = Ax when the eigenvalues are real and different
  3. Complex Eigenvalues (coming soon) — solve linear systems whose eigenvalues are complex
  4. Repeated Eigenvalues (coming soon) — solve linear systems with a repeated eigenvalue
  5. Phase Portraits (coming soon) — sketch trajectories and classify equilibrium points
  6. Nonlinear Systems and Stability (coming soon) — linearize a nonlinear system near an equilibrium point

Also see: matrices and determinants in Precalculus, Unit 4.

Unit 5: Series Solutions

  1. Power Series Solutions (coming soon) — solve linear equations near an ordinary point with power series
  2. Method of Frobenius (coming soon) — find series solutions near a regular singular point

Also see: power series and Taylor series in Calculus 2, Unit 7.

Unit 6: Fourier Series and Partial Differential Equations

  1. Boundary Value Problems (coming soon) — solve two-point boundary value problems and eigenvalue problems
  2. Fourier Series (coming soon) — write a periodic function as a sum of sines and cosines
  3. Fourier Sine and Cosine Series (coming soon) — use even and odd extensions on half intervals
  4. Heat Equation (coming soon) — solve the heat equation with separation of variables
  5. Wave Equation (coming soon) — model a vibrating string
  6. Laplace’s Equation (coming soon) — find steady-state temperatures in a region

For Teachers: Standards Alignment

Lessons follow a typical one-semester college course in ordinary differential equations, with an introduction to Fourier series and partial differential equations at the end. There is no AP exam for this course. The differential equations in AP Calculus AB and BC (slope fields, Euler’s method, separable equations and logistic growth) are taught in Calculus 2, Unit 4.

Previous course: Linear Algebra (coming soon) · Also see: Calculus 3 Index