/*! 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} Q32E Given that the function聽f(x)=x ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Q32E (page 327)

Given that the functionf(x)=x is a solution to y'''-x2y'+xy=0, show that the substitutiony(x)=v(x)f(x)=v(x)x reduces this equation to,xw''+3w'-x3w=0 wherew=v'.

Short Answer

Expert verified

Thus, it is proved that the given equation can be reduced toxw''+3w'-x3w=0.

Step by step solution

01

Use the given functions to reduce the given equation to xw''+3w'-x3w=0

Given thatf(x)=x is a solution toy'''-x2y'+xy=0鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌�......(1)

And

y(x)=v(x)f(x)=v(x)x

Now find the derivative of y for equation (1),

y=vxy'=v+xv'

Use the valuew=v' in the above expression,

y'=v+xwy''=v'+xw'+wy''=w+xw'+wy''=2w+xw'y'''=2w'+w'+xw''y'''=3w'+xw''

02

Conclusion

Substitute the all values in the equation (1),

y'''-x2y'+xy=03w'+xw''-x2(v+xw)+x(vx)=03w'+xw''-x2v-x3w+x2v=03w'+xw''-x3w=0xw''+3w'-x3w=0

Thus, it is proved that the given equation can be reduced toxw''+3w'-x3w=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

Given that{ex,e-x,e2x}is a fundamental solution set for the homogeneous equation corresponding to the equationy'''-2y''-y'+2y=g(x),

determine a formula involving integrals for a particular solution.

use the annihilator method to determinethe form of a particular solution for the given equation. y''-6y'+10y=e3x-x

find a differential operator that annihilates the given function.

xe3xcos5x

Solve the given initial value problem

y'''鈭�4y''+7y'鈭�6y=0y(0)=1y'(0)=0y''(0)=0

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

d2x/dt2-x+y=0x+d2y/dt2-y=e3t
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

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.