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17: Second-Order Differential Equations

  • Page ID
    2625
    • Gilbert Strang & Edwin “Jed” Herman
    • OpenStax
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    We have already studied the basics of differential equations, including separable first-order equations. In this chapter, we go a little further and look at second-order equations, which are equations containing second derivatives of the dependent variable. The solution methods we examine are different from those discussed earlier, and the solutions tend to involve trigonometric functions as well as exponential functions. Here we concentrate primarily on second-order equations with constant coefficients.

    • 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. The solution methods we examine are different from those discussed earlier, and the solutions tend to involve trigonometric functions as well as exponential functions. Here we concentrate primarily on second-order equations with constant coefficients.
    • 17.1: Second-Order Linear Equations
      We often want to find a function (or functions) that satisfies the differential equation. The technique we use to find these solutions varies, depending on the form of the differential equation with which we are working. Second-order differential equations have several important characteristics that can help us determine which solution method to use. In this section, we examine some of these characteristics and the associated terminology.
    • 17.2: Nonhomogeneous Linear Equations
      In this section, we examine how to solve nonhomogeneous differential equations. The terminology and methods are different from those we used for homogeneous equations, so let’s start by defining some new terms.
    • 17.3: Applications of Second-Order Differential Equations
      Scond-order linear differential equations are used to model many situations in physics and engineering. Here, we look at how this works for systems of an object with mass attached to a vertical spring and an electric circuit containing a resistor, an inductor, and a capacitor connected in series. Models such as these can be used to approximate other more complicated situations; e.g., bonds between atoms or molecules are often modeled as springs that vibrate.
    • 17.4: Series Solutions of Differential Equations
      In some cases, power series representations of functions and their derivatives can be used to find solutions to differential equations.
    • 17.R: Chapter 17 Review Exercises

    Thumbnail: A solution to the 2D wavefunction. (CC SA_BY 3.0 Internation; BrentHFoster).


    This page titled 17: Second-Order Differential Equations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.