/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q7 E In Problems 3鈥�8, determine whe... [FREE SOLUTION] | 91影视

91影视

Find study content
Learning Materials

Discover learning materials by subject, university or textbook.

Explanations

Textbooks
All Subjects

Anthropology

Archaeology

Architecture

Art and Design

Bengali

Biology

Business Studies

Greek

Chemistry

Chinese

Combined Science

Computer Science

Economics

Engineering

English

English Literature

Environmental Science

French

Geography

German

History

Hospitality and Tourism

Human Geography

Italian

Japanese

Law

Macroeconomics

Marketing

Math

Media Studies

Medicine

Microeconomics

Music

Nursing

Nutrition and Food Science

Physics

Polish

Politics

Psychology

Religious Studies

Sociology

Spanish

Sports Sciences

Translation

All Subjects

Biology

Business Studies

Chemistry

Combined Science

Computer Science

Economics

English

Environmental Science

Geography

History

Math

Physics

Psychology

Sociology
Features
Features

Discover all of these amazing features with a free account.

Flashcards

91影视 AI

Notes

Study Plans

Study Sets
What鈥檚 new?

Flashcards
Study your flashcards with three learning modes.

Study Sets
All of your learning materials stored in one place.

Notes
Create and edit notes or documents.

Study Plans
Organise your studies and prepare for exams.
91影视
Discover

All the hacks around your studies and career - in one place.

Find a degree

Magazine
Featured

Magazine
Trusted advice for anyone who wants to ace their studies & career.

The largest student job board with the most exciting opportunities.

Verified student deals from top brands.

Discover our mobile app to take your studies anywhere.

Learning Materials

Features

Discover

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 1: Q7 E (page 14)

In Problems 3鈥�8, determine whether the given function is a solution to the given differential equation.
$y = e^{2 x} - 3 e^{- x}$ , $\frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} - 2 y = 0$

Short Answer

Expert verified

The given function is a solution to the given differential equation.

Step by step solution

01

Differentiating the given equation w.r.t. (with respect to) x

Firstly, we will differentiate $y = e^{2 x} - 3 e^{- x}$ with respect to x,

$\frac{dy}{dx} = 2 e^{2 x} + 3 e^{- x}$

Again, differentiating the given function with respect to x,

$\frac{d^{2} y}{{dx}^{2}} = 4 e^{2 x} - 3 e^{- x}$

02

Simplification

Putting the values from step 1 in the L.H.S. (Left-hand side) of the given differential equation,

$\frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} - 2 y = 4 e^{2 x} - 3 e^{- x} - (2 e^{2 x} + 3 e^{- x}) - 2 (e^{2 x} - 3 e^{- x}) \frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} - 2 y = 4 e^{2 x} - 3 e^{- x} - 2 e^{2 x} - 3 e^{- x} - 2 e^{2 x} + 6 e^{- x} \frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} - 2 y = 0$

which is the same as the R.HS. (Right-hand side) of the given differential equation.

Hence, $y = e^{2 x} - 3 e^{- x}$ is a solution to the differential equation $\frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} - 2 y = 0$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Decide whether the statement made is True or False. The function $x (t) = t^{- 3} \sin t + 4 t^{- 3}$ is a solution to $t^{3} \frac{dx}{dt} = \cos t - 3 t^{2} x$ .

Consider the question of Example 5 $y \frac{dy}{dx} - 4 x = 0$

Does Theorem 1 imply the existence of a unique solution to (13) that satisfies $y (x_{0}) = 0$ ?
Show that when $x_{0} 鈮� 0$ equation (13) can鈥檛 possibly have a solution in a neighbourhood of $x = x_{0}$ that satisfies $y (x_{0}) = 0$ .
Show two distinct solutions to (13) satisfying $y (0) = 0$ ( See Figure 1.4 on page 9).

A model for the velocity v at time tof a certain object falling under the influence of gravity in a viscous medium is given by the equation $\frac{dv}{dt} = 1 - \frac{v}{8}$ .From the direction field shown in Figure 1.14, sketch the solutions with the initial conditions v(0) = 5, 8, and 15. Why is the value v = 8 called the 鈥渢erminal velocity鈥�?

Figure 1.14

In Problems 14鈥�24, you will need a computer and a programmed version of the vectorized classical fourth-order Runge鈥揔utta algorithm. (At the instructor鈥檚 discretion, other algorithms may be used.)鈥�

Using the vectorized Runge鈥揔utta algorithm, approximate the solution to the initial value problem $y'' = t^{2} + y^{2}; y (0) = 1, y' (0) = 0$ at $t = 1$ . Starting with $h = 1$ , continue halving the step size until two successive approximations of both $y (1)$ and $y' (1)$ differ by at most 0.1.

Mixing.Suppose a brine containing 0.2 kg of salt per liter runs into a tank initially filled with 500 L of water containing 5 kg of salt. The brine enters the tank at a rate of 5 L/min. The mixture, kept uniform by stirring, is flowing out at the rate of 5 L/min (see Figure 2.6).

(a)Find the concentration, in kilograms per liter, of salt in the tank after 10 min. [Hint:LetAdenote the number of kilograms of salt in the tank attminutes after the process begins and use the fact that

rate of increase inA=rate of input- rate of exit.

A further discussion of mixing problems is given in Section 3.2.]

(b)After 10 min, a leak develops in the tank and an additional liter per minute of mixture flows out of the tank (see Figure 2.7). What will be the concentration, in kilograms per liter, of salt in the tank 20 min after the leak develops? [Hint:Use the method discussed in Problems 31 and 32.]

See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.