2: Applications of First Order Equations

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In this chapter, we consider applications of first order differential equations.

2.1: Growth and Decay
This page covers exponential growth and decay, focusing on radioactive decay and continuous compounding of interest. It introduces differential equations, emphasizing that the change rate is proportional to size, and defines half-life in radioactive contexts. The mathematical model illustrates continuous growth with an annual rate of $2,600, including a comparison of predicted versus exact values over a decade.
- 2.1E: Growth and Decay (Exercises)
2.2: Cooling and Mixing
This page covers Newton's Law of Cooling, explaining how an object's temperature changes relative to its environment, represented by an exponential decay model. It provides examples of cooling objects and introduces mixing problems, particularly concerning the salt concentration in a tank. Here, differential equations are key for modeling salt quantity as water is added and drained. The text concludes with solutions for the salt amount $Q(t)$ and concentration $K(t)$ over time.
- 2.2E: Cooling and Mixing (Exercises)
2.3: Elementary Mechanics
This page discusses Newton's Second Law of Motion, illustrating its application to forces acting on falling objects subject to gravity and air resistance. It covers concepts like terminal velocity for masses with different initial conditions and the dynamics of a mass in free fall. Additionally, it explores escape velocity for space vehicles, explaining how gravitational force affects their motion post-fuel burnout.
- 2.3E: Elementary Mechanics (Exercises)
2.4: Autonomous Second Order Equations
This page discusses autonomous second-order differential equations, particularly in the context of spring-mass and pendulum systems. It emphasizes the transformation to first-order equations, stability and uniqueness of solutions, and the nature of trajectories in phase planes. The text covers oscillatory behavior and stability conditions for undamped and damped systems, highlighting specific examples.
- 2.4E: Autonomous Second Order Equations (Exercises)
2.5: Applications to Curves
This page covers the generation of families of curves through parameter variation in equations such as $y - cx^2 = 0$ for parabolas and $y = x + c$ for lines. It explains deriving differential equations from these families, implicit differentiation, and finding tangent lines.
- 2.5E: Applications to Curves (Exercises)

Thumbnail: False color time-lapse video of E. coli colony growing on microscope slide. This growth can be model with first order logistic equation. Added approximate scale bar based on the approximate length of 2.0 μm of E. coli bacteria. (CC BY-SA 4.0 International; Stewart EJ, Madden R, Paul G, Taddei F).

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