# 4: Applications of First Order Equations

- Page ID
- 30710

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

- 4.1: Growth and Decay
- This section begins with a discussion of exponential growth and decay, which you have probably already seen in calculus. We consider applications to radioactive decay, carbon dating, and compound interest. We also consider more complicated problems where the rate of change of a quantity is in part proportional to the magnitude of the quantity, but is also influenced by other other factors for example, a radioactive susbstance is manufactured at a certain rate, but decays at a rate proportional

- 4.2: Cooling and Mixing
- This section deals with applications of Newton's law of cooling and with mixing problems.

- 4.3: Elementary Mechanics
- This section discusses applications to elementary mechanics involving Newton's second law of motion. The problems considered include motion under the influence of gravity in a resistive medium, and determining the initial velocity required to launch a satellite.

- 4.4: Autonomous Second Order Equations
- This section deals with methods for dealing with a type of second order equation that often arises in applications of Newton's second law of motion, by reformulating it as first order equation with a different independent variable. Although the method doesn't usually lead to an explicit solution of the given equation, it does provide valuable insights into the behavior of the solutions.

- 4.5: Applications to Curves
- This section deals with applications of differential equations to curves.

*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. Image used with permission (CC BY-SA 4.0 International; Stewart EJ, Madden R, Paul G, Taddei F).*