Unit 3 of 13

First-Order Models and Qualitative Behavior

  • 5 lessons
  • Videos 14 to 18

First-order equations describe growth, cooling, mixing and motion. This unit reads stability from a phase line and builds models whose solutions are checked against their units and assumptions.

For an autonomous equation, the sign of the rate between equilibria shows which ones attract, which repel and which attract from one side only. Exponential decay gives the half-life and the time constant, and Newton's law of cooling follows the difference from a constant surrounding temperature. The logistic model adds a carrying capacity, and constant harvesting creates a threshold below which the population declines. Mixing problems keep amount and concentration apart, with a changing volume or two coupled tanks. Falling objects under linear and quadratic drag approach a terminal velocity.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Phase Lines and Stability

  1. Autonomous Equations and Phase-Line Stability

    How to read the long-term behavior of an autonomous equation from the sign of its rate, with equilibrium solutions, the phase line, stability, asymptotic stability, semistability and the derivative test.

    Video 14

Growth, Decay and Population Models

  1. Exponential Growth, Decay, and Newton Cooling

    How a rate proportional to the amount gives exponential growth and decay, what the half-life and the time constant measure, and how Newton's law of cooling moves a temperature toward a constant ambient temperature.

    Video 15

  2. Logistic Growth and Harvesting

    How the logistic model limits growth at a carrying capacity, where natural growth is fastest, and how constant harvesting creates a threshold equilibrium, a critical harvesting rate and a boundary of the physical domain.

    Video 16

Mixing and Motion Models

  1. Mixing and Compartment Models

    How a mass balance for the amount of a substance gives a differential equation, why perfect mixing ties the outflow to the concentration, and how variable volume and coupled compartments enter the model.

    Video 17

  2. Motion with Drag and Terminal Velocity

    How a chosen positive direction fixes the signs of gravity and drag, how linear and quadratic drag lead to first-order equations for velocity, and how the transient velocity decays toward the terminal velocity.

    Video 18