Calculus: Early Transcendentals, 2nd Edition

2nd Edition

ISBN: 9780321965165

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Textbook Question

Chapter D1.2, Problem 38E

Equilibrium solutions A differential equation of the form y′(t) = f(y) is said to be autonomous (the function f depends only on y). The constant function y = y₀ is an equilibrium solution of the equation provided f(y₀) = 0 (because then y′(t) = 0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

a. Find the equilibrium solutions.
b. Sketch the direction field, for t ≥ 0.
c. Sketch the solution curve that corresponds to the initial condition y (0) = 1.

38. y′(t) = 2y + 4

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter D1.2, Problem 37E

Chapter D1.2, Problem 39E

Students have asked these similar questions

y" +p (t)y' +q(t)y = 0 (i), can be put in more suitable form for finding a solution by making a change of the independent and/or dependent variables. Determine conditions on p and q that enable this equation to be transformed into an equation with constant coefficients by a change of the independent variable. Let r = u (t) be the new independent variable. It is easy to show (島) ( + ( +p(t) + a()y = 0 (1i). dy that dt dz dy d'y and 2 d'y d²z dy dt dz The differential equation becomes dt dr dt d'y d'z de dz dy d'y , 9 and y must be proportional. If g (t) > 0, then choose the constant of proportionality to be 1. Hence, x = u (t) = [g (t)]i dt. With a chosen this way, the coefficient of in equation (ii) is also a constant, provided that the function H = g'(t) +2p(t)q(t) is a dy In order for equation (ii) to have constant coefficients, the coefficients of constant. Thus, equation (i) can be transformed into an equation with constant coefficients by a change of the independent variable,…

A force of 4 pounds stretches a spring 1 foot. A mass weighing 3.2 pounds is attached to the spring, and the system is then immersed in a medium that offers a damping force numerically equal to 0.4 times the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from rest from a point 1 foot above the equilibrium position. x(t) = ft (b) Express the equation of motion in the form x(t) = Ae-¹t sin(√w² - 2²t + p), which is given in (23) of Section 3.8. (Round up to two decimal places.) x(t) = ft (c) Find the first time at which the mass passes through the equilibrium position heading upward. (Round your answer to three decimal places.) S

Assume that N(t) denotes the density of an insect species at time t and P(t) denotes the density of its predator at time t. The insect species is an agricultural pest, and its predator is used as a biological control agent. Their dynamics are given below by the system of differential equations. Complete parts (a) through (c). dN = 7N - 5PN dt dP = 4PN - P dt ..... (a) Explain why dN = 7N describes the dynamics of the insect in the absence of the predator. dt If there are no predators present, then P(t) = for all t. Substitute P = in the given differential dN equations to get dt So in the absence of the predators, the above equation describes the dynamics of the insect population. dN Solve the equation, dt N(t) = (Type an expression using t as the variable.) Describe what happens to the insect population in the absence of the predator. In the absence of the predator, the insect population

Chapter D1 Solutions

Calculus: Early Transcendentals, 2nd Edition

Ch. D1.1 - Prob. 1E Ch. D1.1 - Prob. 2E Ch. D1.1 - Prob. 3E Ch. D1.1 - If the general solution of a differential equation...Ch. D1.1 - Does the function y(t) = 2t satisfy the...Ch. D1.1 - Does the function y(t) = 6e3t satisfy the initial...Ch. D1.1 - Verifying general solutions Verify that the given...Ch. D1.1 - Verifying general solutions Verify that the given...Ch. D1.1 - Verifying general solutions Verify that the given...Ch. D1.1 - Verifying general solutions Verify that the given...

Additional Math Textbook Solutions

Find more solutions based on key concepts

In Exercises 13–16, find the margin of error for the values of c, ?, and n. 13. c = 0.95, ? = 5.2, n = 30

Elementary Statistics: Picturing the World (7th Edition)

Explain the meaning of the term “statistically significant difference” in statistics terminology.

Intro Stats, Books a la Carte Edition (5th Edition)

Critical Values. In Exercises 41–44, find the indicated critical value. Round results to two decimal places. 42...

Elementary Statistics (13th Edition)

Continuity at a point Determine whether the following functions are continuous at a. Use the continuity checkli...

Calculus: Early Transcendentals (2nd Edition)

The equivalent expression of x(y+z) by using the commutative property.

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Mathematical Connections Explain why a number and a numeral are considered different.

A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Solution Summary: The author explains that the equilibrium solution of the given differential equation is y(t)=2y+4.

Concept explainers

Videos

Want to see the full answer?

Chapter D1 Solutions

Additional Math Textbook Solutions