Ch. 7 Key Terms - Calculus Volume 3 | OpenStax

Skip to Content Go to accessibility page

Key Terms

boundary conditions: the conditions that give the state of a system at different times, such as the position of a spring-mass system at two different times

boundary-value problem: a differential equation with associated boundary conditions

characteristic equation: the equation $a λ^{2} + b λ + c = 0$ for the differential equation $a y^{″} + b y^{'} + c y = 0$

complementary equation: for the nonhomogeneous linear differential equation

$a_{2} (x) y^{″} + a_{1} (x) y^{'} + a_{0} (x) y = r (x),$

the associated homogeneous equation, called the complementary equation, is

$a_{2} (x) y^{″} + a_{1} (x) y^{'} + a_{0} (x) y = 0$

homogeneous linear equation: a second-order differential equation that can be written in the form $a_{2} (x) y^{″} + a_{1} (x) y^{'} + a_{0} (x) y = r (x),$ but $r (x) = 0$ for every value of $x$

linearly dependent: a set of functions $f_{1} (x), f_{2} (x),…, f_{n} (x)$ for which there are constants $c_{1}, c_{2},… c_{n},$ not all zero, such that $c_{1} f_{1} (x) + c_{2} f_{2} (x) + ⋯ + c_{n} f_{n} (x) = 0$ for all x in the interval of interest

linearly independent: a set of functions $f_{1} (x), f_{2} (x),…, f_{n} (x)$ for which there are no constants $c_{1}, c_{2},… c_{n},$ such that $c_{1} f_{1} (x) + c_{2} f_{2} (x) + ⋯ + c_{n} f_{n} (x) = 0$ for all x in the interval of interest

method of undetermined coefficients: a method that involves making a guess about the form of the particular solution, then solving for the coefficients in the guess

method of variation of parameters: a method that involves looking for particular solutions in the form $y_{p} (x) = u (x) y_{1} (x) + v (x) y_{2} (x),$ where $y_{1}$ and $y_{2}$ are linearly independent solutions to the complementary equations, and then solving a system of equations to find $u (x)$ and $v (x)$

nonhomogeneous linear equation: a second-order differential equation that can be written in the form $a_{2} (x) y^{″} + a_{1} (x) y^{'} + a_{0} (x) y = r (x),$ but $r (x) \neq 0$ for some value of $x$

particular solution: a solution $y_{p} (x)$ of a differential equation that contains no arbitrary constants

RLC series circuit: a complete electrical path consisting of a resistor, an inductor, and a capacitor; a second-order, constant-coefficient differential equation can be used to model the charge on the capacitor in an RLC series circuit

simple harmonic motion: motion described by the equation $x (t) = c_{1} cos (ω t) + c_{2} sin (ω t),$ as exhibited by an undamped spring-mass system in which the mass continues to oscillate indefinitely

steady-state solution: a solution to a nonhomogeneous differential equation related to the forcing function; in the long term, the solution approaches the steady-state solution

Order a print copy

Citation/Attribution

Reuse and redistribution of this content in digital or print format:

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's prior written permission.
This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License, which means that you can reuse and modify the material only for noncommercial purposes, must attribute OpenStax, and must distribute any derivative works under the same license.
Any commercial printing of this textbook, including using a local or custom printer, must be approved by OpenStax, and proper citation provided.
OpenStax-copyrighted images, activities, assessments, and similar components of this book are subject to the same licensing – CC-BY-NC-SA. They can be used for noncommercial purposes with attribution. Commercial use requires permission.
Permission requests: Anyone who intends to incorporate this content (including text, images, and other components) into large language models, use it in AI offerings, use it commercially (including in print), and/or has questions about another use case is welcome to complete our reuse request form.

Attribution information

If you are redistributing all or part of this book in a noncommercial print format, then you must include on every physical page the following attribution:
Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction
If you are redistributing all or part of this book in a noncommercial digital format, then for every page that includes OpenStax content, you must license the derivative work under the same CC-BY-NC-SA license as the original, and include on every digital page view the following attribution:
Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction

Citation information

The information below includes the information needed to generate citations in most major styles (APA, MLA, etc.); you must reformat and organize the information as needed to fit the requirements of the style. Use the information below to generate a citation. We recommend using a citation tool such as this one.

Authors: Gilbert Strang, Edwin “Jed” Herman
Publisher/website: OpenStax
Book title: Calculus Volume 3
Publication date: Mar 30, 2016
Location: Houston, Texas
Book URL: https://openstax.org/books/calculus-volume-3/pages/1-introduction
Section URL: https://openstax.org/books/calculus-volume-3/pages/7-key-terms

© Jul 15, 2026 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo, and Rice University name, and Rice University logo trademarks, or wordmarks are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.