Find study content
Learning Materials

Discover learning materials by subject, university or textbook.

Explanations

Textbooks

Exams
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

EXAM TYPES

GCSE

IGCSE

AS

A Level

International A Level

University Admissions Tests

GCSE SUBJECTS

GCSE 1

GCSE 2

GCSE 3

GCSE 4

GCSE 5

SOME-TEXT

IGCSE SUBJECTS

IGCSE 1

AS SUBJECTS

AS 1

A Level SUBJECTS

A Level 1

International A Level SUBJECTS

International A Level 1

University Admissions Tests SUBJECTS

University Admissions Tests 1
Features
Features

Discover all of these amazing features with a free account.

Flashcards

Vaia AI

Notes

Study Plans

Study Sets
What’s 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.
Resources
Discover

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

Find a job

Find a degree

Student Deals

Magazine

Mobile App
Featured

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

Job Board
The largest student job board with the most exciting opportunities.

Vaia Deals
Verified student deals from top brands.

Our App
Discover our mobile app to take your studies anywhere.

Go to App

Learning Materials

Features

Discover

Fundamentals Of Differential Equations And Boundary Value Problems

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

$Math Studyset Vaia Explanations$ Math

9th Edition · 616 pages

ISBN: 9780321977069

Chapter 6: Q30E (page 337)

use the annihilator method to determinethe form of a particular solution for the given equation.
$y''' + 2 y'' - y' - 2 y = e^{x} - 1$

Short Answer

Expert verified

$y_{p} (x) = c_{1} + c_{5} x e^{x}$

Step by step solution

01

Solve the homogeneous of the given equation

The homogeneous of the given equation is

$(D^{3} + 2 D^{2} - D - 2) [y] = 0$

The solution of the homogeneous is

$y_{h} (x) = c_{1} e^{- 2 x} + c_{2} e^{- x} + c_{3} e^{x}$ (1)

Now $e^{x} - 1$ is annihilated by $(D^{2} - D)$

Then, every solution to the given nonhomogeneous equation also satisfies

. $(D^{2} - D)$ $(D^{3} + 2 D^{2} - D - 2) [y] = 0$

Then

$y (x) = c_{1} + c_{2} e^{- 2 x} + c_{3} e^{- x} + c_{4} e^{x} + c_{5} x e^{x}$ (2)

is the general solution to this homogeneous equation

We know $u (x) = u_{h} + u_{p}$

Comparing (1) & (2)

$y_{p} (x) = c_{1} + c_{5} x e^{x}$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

As an alternative proof that the functions $e^{r_{1} x}, e^{r_{2} x}, e^{r_{3} x}, ..., e^{r_{n} x}$ are linearly independent on (∞,-∞) when $r_{1}, r_{2}, . .. r_{n}$ are distinct, assume $C_{1} e^{r_{1} x} + C_{2} e^{r_{2} x} + C_{3} e^{r_{3} x} + . .. + C_{n} e^{r_{n} x}$ holds for all x in (∞,-∞) and proceed as follows:

(a) Because the ri’s are distinct we can (if necessary)relabel them so that $r_{1} > r_{2} > r_{3} > . .. > r_{n}$ .Divide equation (33) by to obtain $C_{1} + \frac{C_{2} e^{r_{2} x}}{e^{r_{2} x}} + \frac{C_{3} e^{r_{3} x}}{e^{r_{3} x}} + . .. + \frac{C_{n} e^{r_{n} x}}{e^{r_{n} x}} = 0$ Now let x→∞ on the left-hand side to obtainC1 = 0.(b) Since C1 = 0, equation (33) becomes

$C_{2} e^{r_{2} x} + C_{3} e^{r_{3} x} + . .. + C_{n} e^{r_{n} x}$ = 0for all x in(∞,-∞). Divide this equation by $e^{r_{2} x}$

and let x→∞ to conclude that C2 = 0.

(c) Continuing in the manner of (b), argue that all thecoefficients, C1, C2, . . . ,Cn are zero and hence $e^{r_{1} x}, e^{r_{2} x}, e^{r_{3} x}, . .., e^{r_{n} x}$ are linearly independent on(∞,-∞).

use the method of undetermined coefficients to determine the form of a particular solution for the given equation.

$y''' + 3 y'' - 4 y = e^{- 2 x}$

use the method of undetermined coefficients to determine the form of a particular solution for the given equation.

$y''' + y'' - 5 y' + 3 y = e^{- x} + s i n x$

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

${\cos 2 x, \cos^{2} x, \sin^{2} x}$ on $(- \infty, \infty)$

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.

$x (x + 1) y''' - 3 xy' + y = 0 y (\frac{- 1}{2}) = 1, y' (\frac{- 1}{2}) = y'' (\frac{- 1}{2}) = 0$

See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.

Sign-up for free