/*! 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} Q6E Find a general solution for the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Q6E (page 332)

Find a general solution for the differential equation with x as the independent variable:

y'''鈭�y''+2y=0

Short Answer

Expert verified

The general solution for the differential equation with x as the independent variableis .y(x)=c1e鈭�x+c2e鈭�5x+c3e4x

Step by step solution

01

Auxiliary equation:

The given differential equationis y'''鈭�y''+2y=0. To solve this equation, we look at its auxiliary equation which is m3鈭�m2+2=0 . Observe that -1 is a solution of this equation. So,

m3鈭�m2+2=(m+1)(m2鈭�2m+2)

02

Inspecting the sum further:

To get the other two roots of auxillary equation, we need to solve m3鈭�m2+2=0. We have,

m=2卤4鈭�82=1卤i

03

General solution:

We have m = -1,1卤i.From (7) of 328 and (18) of page 330, we conclude that the general solution of the given differential equation is y=C1e鈭�x+C2excosx+C3exsinxwhere C1,C2,C3 are arbitrary constants.

The solution of the given differential equation isy=C1e鈭�x+C2excosx+C3exsinx , where C1,C2,C3 are arbitrary constant.

Hence the final solution is y(x)=c1e鈭�x+c2e鈭�5x+c3e4x

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

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.

xx+1y'''-y'+xy=0y12=y'12=-1,鈥�鈥�y''12=1

In Problems 38 and 39, use the elimination method of Sectionto find a general solution to the given system.

x-d2y/dt2=t+1

dx/dt+dy/dt-2y=et

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

{x,鈥�x2,鈥�x3,鈥�x4}on(-鈭�,鈥�鈭�)

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

{x-1,鈥�x12,鈥�x}on(0,鈥�鈭�)

Let y1x2= Cerx, where C (鈮�0) and r are real numbers,be a solution to a differential equation. Supposewe cannot determine r exactly but can only approximateit by r. Let (x) =Cerxand consider the error

|y(x)鈭�y~(x)|

(a) If r andr~are positive, r 鈮犅� , show that the errorgrows exponentially large as x approaches + 鈭�.

(b) If r andrare negative, r鈮� , show that the errorgoes to zero exponentially as x approaches + 鈭�.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

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.