Module 3: Applications of First-Order ODEs
See first-order differential equations in action through real-world models: exponential growth and decay, population dynamics, mixing tanks, and Newton's law of cooling.
Your Progress
Learning Objectives
By the end of this module, you will be able to:
- Model exponential growth and decay with dP/dt = kP and find half-life or doubling time
- Analyze the logistic equation dP/dt = rP(1 - P/K) including equilibria and carrying capacity
- Set up mixing problems using rate-in minus rate-out and solve the resulting linear ODE
- Apply Newton's law of cooling dT/dt = k(T - T_env) to forensic and practical problems
- Translate word problems into differential equations and interpret solutions in context
Module Lessons
Exponential Growth and Decay
Model populations, radioactive decay, and compound interest with dP/dt = kP. Learn half-life and doubling time.
35-45 minutes
Population Models: The Logistic Equation
Go beyond exponential growth with the logistic model, carrying capacity, equilibria, and harvesting.
35-45 minutes
Mixing Problems
Set up and solve ODEs for tanks with inflow and outflow of solutions. Master the rate-in minus rate-out framework.
30-40 minutes
Newton's Law of Cooling
Model temperature change with dT/dt = k(T - T_env). Apply to forensics, cooking, and engineering.
30-40 minutes
After the Lessons
Practice Problems
Apply what you learned with 10 problems covering growth/decay, logistic models, mixing, and cooling.
Practice ProblemsModule Quiz
Test your understanding with a 10-question quiz. Detailed solutions included!
Take Module QuizStudy Materials
Printable study guide and quick reference card for exam prep.