Unit 1 of 13

Language, Solutions and Modeling

  • 6 lessons
  • Videos 1 to 6

A differential equation relates an unknown function to its derivatives. This unit sets up the vocabulary, the meaning of a solution, the calculus it relies on and the first existence questions.

Equations are classified as ordinary or partial, by order and as linear or nonlinear, and two meanings of homogeneous are kept apart. A solution may be explicit or implicit and is checked against the equation, the data and its domain. A review of calculus and complex numbers leads to Euler's formula, and direction fields show solution curves without a formula. If the right-hand side is continuous and locally Lipschitz in the unknown, an initial value selects one solution on a maximal interval, which can end in finite-time blowup. A balance law with consistent units gives a first model.

Every lesson is one video. Each lesson page has the definitions and an example for every key term.

Equations, Solutions and Calculus Tools

  1. Differential Equations and Their Language

    What a differential equation and its unknown function are, how ordinary and partial equations differ, and how to read order, dependent and independent variables, linearity and the meanings of homogeneous.

    Video 1

  2. Solutions, Initial Conditions, and Boundary Conditions

    What explicit, implicit, general and particular solutions are, how to verify a solution against its equation, data and domain, and how initial-value and boundary-value problems select solutions.

    Video 2

  3. Calculus and Complex Numbers for Differential Equations

    The product and chain rules, antiderivatives and constants of integration, the fundamental theorem of calculus, exponentials and logarithms, complex numbers and Euler's formula, used to build and check solutions.

    Video 3

Direction Fields, Existence and Uniqueness

  1. Direction Fields and Solution Curves

    How a first-order equation gives a local slope at each point, how those slopes form a direction field organized by isoclines, and how solution curves and a solution family fit the field exactly.

    Video 4

  2. Existence, Uniqueness, and Maximal Intervals

    When an initial-value problem has a local solution and when it has exactly one, what the local Lipschitz condition adds to continuity, and how maximal intervals, finite-time blowup and nonuniqueness arise.

    Video 5

Modeling and Units

  1. Modeling, Units, and Nondimensionalization

    How to choose a state and parameters, write a balance law for a well-mixed tank, check units and dimensional consistency, and rescale by characteristic scales into a nondimensional equation.

    Video 6