Skip to ContentGo to accessibility page
OpenStax Logo
Calculus Volume 2

Key Terms

Calculus Volume 2Key Terms

Key Terms

asymptotically semi-stable solution
y=ky=k if it is neither asymptotically stable nor asymptotically unstable
asymptotically stable solution
y=ky=k if there exists ε>0ε>0 such that for any value c∈(k−ε,k+ε)c∈(k−ε,k+ε) the solution to the initial-value problem y′=f(x,y),y(x0)=cy′=f(x,y),y(x0)=c approaches kk as xx approaches infinity
asymptotically unstable solution
y=ky=k if there exists ε>0ε>0 such that for any value c∈(k−ε,k+ε)c∈(k−ε,k+ε) the solution to the initial-value problem y′=f(x,y),y(x0)=cy′=f(x,y),y(x0)=c never approaches kk as xx approaches infinity
autonomous differential equation
an equation in which the right-hand side is a function of yy alone
carrying capacity
the maximum population of an organism that the environment can sustain indefinitely
differential equation
an equation involving a function y=y(x)y=y(x) and one or more of its derivatives
direction field (slope field)
a mathematical object used to graphically represent solutions to a first-order differential equation; at each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point
equilibrium solution
any solution to the differential equation of the form y=c,y=c, where cc is a constant
Euler’s Method
a numerical technique used to approximate solutions to an initial-value problem
general solution (or family of solutions)
the entire set of solutions to a given differential equation
growth rate
the constant r>0r>0 in the exponential growth function P(t)=P0ertP(t)=P0ert
initial population
the population at time t=0t=0
initial value(s)
a value or set of values that a solution of a differential equation satisfies for a fixed value of the independent variable
initial velocity
the velocity at time t=0t=0
initial-value problem
a differential equation together with an initial value or values
integrating factor
any function f(x)f(x) that is multiplied on both sides of a differential equation to make the side involving the unknown function equal to the derivative of a product of two functions
linear
description of a first-order differential equation that can be written in the form a(x)y′+b(x)y=c(x)a(x)y′+b(x)y=c(x)
logistic differential equation
a differential equation that incorporates the carrying capacity KK and growth rate rr into a population model
order of a differential equation
the highest order of any derivative of the unknown function that appears in the equation
particular solution
member of a family of solutions to a differential equation that satisfies a particular initial condition
phase line
a visual representation of the behavior of solutions to an autonomous differential equation subject to various initial conditions
separable differential equation
any equation that can be written in the form y′=f(x)g(y)y′=f(x)g(y)
separation of variables
a method used to solve a separable differential equation
solution curve
a curve graphed in a direction field that corresponds to the solution to the initial-value problem passing through a given point in the direction field
solution to a differential equation
a function y=f(x)y=f(x) that satisfies a given differential equation
standard form
the form of a first-order linear differential equation obtained by writing the differential equation in the form y′+p(x)y=q(x)y′+p(x)y=q(x)
step size
the increment hh that is added to the xx value at each step in Euler’s Method
threshold population
the minimum population that is necessary for a species to survive
PreviousNext
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-2/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-2/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.

© 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.