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7.9 Logistic Models with Differential Equations

Part of Differential Equations in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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A logistic model describes growth that begins rapidly but slows as the quantity approaches a limiting value called the carrying capacity. In AP Calculus, the logistic differential equation is commonly written in a form like
dydt=ky(1−yL)\frac{dy}{dt}=ky\left(1-\frac{y}{L}\right)
where L is the carrying capacity. When y is small, growth is approximately exponential, but as y gets closer to L, the factor
1−yL1-\frac{y}{L}
gets smaller, so the growth rate decreases. This topic is BC only.
An image of a logistic graph is shown above.

In this unit

9 lessons

  1. 7.1Modeling Situations with Differential EquationsPreview lesson
  2. 7.2Verifying Solutions for Differential EquationsPreview lesson
  3. 7.3Sketching Slope FieldsPreview lesson
  4. 7.4Reasoning Using Slope FieldsPreview lesson
  5. 7.5Finding General Solutions Using Separation of VariablesPreview lesson
  6. 7.6Finding Particular Solutions Using Initial Conditions and Separation of VariablesPreview lesson
  7. 7.7Exponential Models with Differential EquationsPreview lesson
  8. 7.8Approximating Solutions Using Euler’s MethodPreview lesson
  9. 7.9Logistic Models with Differential EquationsThis lesson

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