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Chapter 6: Q37E (page 338)

Use the annihilator method to show that iff(x)in (4) has the formf(x)=acos尾x+bsin尾x,

then equation (4) has a particular solution of the form

(18)yp(x)=xs{Acos尾x+Bsin尾x} ,where sis chosen to be the smallest nonnegative integer such thatx3cos尾xandx3sin尾xare not solutions to the corresponding homogeneous equation

Short Answer

Expert verified

yp=xs(Acos尾x+Bsin尾x)is the form of particular solution.

Step by step solution

01

Definition

A linear differential operatorAis said to annihilate a functionfif A[f](x)=0鈥�鈥�鈥�鈥�鈥�--(2)for all. That is, A annihilates fiffis a solution to the homogeneous linear differential equation (2) on(-鈭�,鈭�).

02

For particular solution

Equation (4) is given byany(n)+an-1yn-1+鈥�鈥�..+a0y=f

Also givenf(x)=acos尾x+bsin尾x

Then,anDn+an-1Dn-1+鈥�鈥�..+a0y=acos尾x+bsin尾x

So,sin尾xcos尾x is annihilated byD2+尾2

So,D2+尾2anDn+an-1Dn-1+鈥�鈥�..+a0y=0

For particular solution i.e;ypcheck ifcos尾x,sin尾xare solutions to homogeneous, particular solution is different than homogeneous solution choose

yp=xs(Acos尾x+Bsin尾x)

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Most popular questions from this chapter

Use the result of Problem 34 to construct a third-order differential equation for which{x,鈥�sinx,鈥�cosx} is a fundamental solution set.

find a differential operator that annihilates the given function.

3x2-6x+1

Find a general solution to the givenhomogeneous equation.(D鈭�1)2(D+3)(D2+2D+5)2[y]=0

find a general solution to the given equation.

y'''+4y''+y'-26y=e-3xsin2x+x

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

y'''+y''-5y'+3y=e-x+sinx
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