Unit 9 of 10

First-Order Differential Equations

  • 7 lessons
  • Parts 229 to 242

A differential equation relates a function to its own rate of change. Population growth, cooling objects and mixing tanks are all described this way, and solving the equation means finding the function.

The unit covers what a solution is, direction fields, separable and linear equations, useful substitutions, and models for logistic growth, mixing, cooling and simple circuits.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Equations and Solutions

  1. Differential Equations and Initial-Value Problems

    What a differential equation and its solutions are, how to check a proposed solution, and why a starting value and an interval are part of the answer.

    Theory part 229Practice part 230

  2. Direction Fields, Equilibria, and Uniqueness

    How to read the behavior of solutions from a slope field or a phase line without solving the equation, how to classify equilibria as stable or unstable, and when an initial-value problem has exactly one solution.

    Theory part 231Practice part 232

Solution Methods

  1. Separable Differential Equations

    How to solve equations of the form y' = g(x)h(y) by separating the variables and integrating, without losing constant solutions, and how to find the interval on which a solution lives.

    Theory part 233Practice part 234

  2. First-Order Linear Differential Equations

    How to solve y' + P(x)y = Q(x) with an integrating factor or by variation of the parameter, and why every such problem with continuous coefficients has exactly one solution on the whole interval.

    Theory part 235Practice part 236

  3. First-Order Substitutions

    How a well chosen change of variable turns Bernoulli equations into linear ones and equations of the form y' = F(y/x) into separable ones, and which solutions and domains need extra care along the way.

    Theory part 237Practice part 238

Modelling with Differential Equations

  1. Logistic Growth and Resource Models

    How the logistic equation models growth that levels off at a carrying capacity, how to solve it and classify its equilibria, and how harvesting changes which yields a population can sustain.

    Theory part 239Practice part 240

  2. Mixing, Cooling, and First-Order Circuits

    How to set up first order models for salt in a tank, a cooling object and simple electric circuits from a balance of rates, solve them, and read off what each constant means in physical units.

    Theory part 241Practice part 242