![]() Here we concentrate primarily on second-order equations with constant coefficients. The solution methods we examine are different from those discussed earlier, and the solutions tend to involve trigonometric functions as well as exponential functions. 17.0: Prelude to Second-Order Differential Equations In this chapter, we look at second-order equations, which are equations containing second derivatives of the dependent variable.Thumbnail: An exponential growth model of population.\) Chapter 8 Review Exercises These are homework exercises to accompany OpenStax's "Calculus" Textmap.Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit. We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value. 8.5: First-order Linear Equations Any first-order linear differential equation can be written in the form y′+p(x)y=q(x).8.4E: Exercises for the Logistic Equation.In this section, we study the logistic differential equation and see how it applies to the study of population dynamics in the context of biology. A more realistic model includes other factors that affect the growth of the population. We saw this in an earlier chapter in the section on exponential growth and decay, which is the simplest model. 8.4: The Logistic Equation Differential equations can be used to represent the size of a population as it varies over time.8.3E: Exercises for Separable Differential Equations.We illustrate a few applications at the end of the section. These equations are common in a wide variety of disciplines, including physics, chemistry, and engineering. 8.3: Separable Differential Equations We now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations.8.2E: Exercises for Direction Fields and Numerical Methods.We will also study numerical methods for solving differential equations, which can be programmed by using various computer languages or even by using a spreadsheet program. 8.2: Direction Fields and Numerical Methods In some cases it is possible to predict properties of a solution to a differential equation without knowing the actual solution.8.1E: Exercises for Basics of Differential Equations.Solving such equations often provides information about how quantities change and frequently provides insight into how and why the changes occur. ![]() Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(x) and its derivative, known as a differential equation.
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