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Chapter 6: Q34E (page 337)

Use the annihilator method to show that ifa0鈮�0in equation (4) and fxhas the form (17) f(x)=bmxm+bm-1xm-1+鈰�+b1x+b0, thenyp(x)=Bmrxm+Bm-1xm-1+鈰�+B1x+B0is the form of a particular solution to equation (4).

Short Answer

Expert verified

yp=Bmxm+鈥�鈥�鈥�+B1x+B0is the form of particular solution.

Step by step solution

01

Definition

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

02

Check for particular solution

It is given thatf(x)=bmxm+鈥�鈥�鈥�+b1x+b0 and a0鈮�0.

Then the ygis given by:

any(n)+鈥�..+a1y'+a0y=f

Soyp=Bmxm+鈥�鈥�鈥�+B1x+B0

(Then yp鈮�yg)

Therefore Homogeneous auxiliary equation is not particular solution for f's, annihilator.

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Most popular questions from this chapter

(a) Derive the form y(x)=A1ex+A2e鈭�x+A3cosx+A4sinx for the general solution to the equation y(4)=y, from the observation that the fourth roots of unity are 1, -1, i, and -i.

(b) Derive the form

y(x)=A1ex+A2e鈭�x/2cos(3x2)+A3e鈭�x/2sin(3x2)

for the general solution to the equation y(3)=yfrom the observation that the cube roots of unity are 1,ei2蟺3 , and e鈭�i2蟺3.

Reduction of Order. If a nontrivial solution f(x) is known for the homogeneous equation

,y(n)+p1(x)y(n-1)+...+pn(x)y=0

the substitutiony(x)=v(x)f(x)can be used to reduce the order of the equation for second-order equations. By completing the following steps, demonstrate the method for the third-order equation

(35)y'''-2y''-5y'+6y=0

given thatf(x)=ex is a solution.

(a) Sety(x)=v(x)exand compute y鈥�, y鈥�, and y鈥�.

(b) Substitute your expressions from (a) into (35) to obtain a second-order equation in.w=v'

(c) Solve the second-order equation in part (b) for w and integrate to find v. Determine two linearly independent choices for v, say, v1and v2.

(d) By part (c), the functions y1(x)=v1(x)exand y2(x)=v2(x)exare two solutions to (35). Verify that the three solutions ex,鈥�y1(x), and y2(x)are linearly independent on(-鈭�,鈥�鈭�)

Find a general solution to the givenhomogeneous equation.(D+1)2(D鈭�6)3(D+5)(D2+1)(D2+4)2[y]=0

In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.

y'''+y'=tanx

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'.
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