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Chapter 6: Q22E (page 337)

use the annihilator method to determinethe form of a particular solution for the given equation.

y''+6y'+8y=e3x-sinx

Short Answer

Expert verified

yp(x)=c3e3x+c4sinx+c5cosx

Step by step solution

01

Solve the homogeneous of the given equation

The homogeneous of the given equation is

D2+6D+8[y]=0鈬�(D+2)(D+4)[y]=0

The solution of the homogeneous is

yh(x)=c1e-2x+c2e-4x (1)

Letg(x)=e3x

Then

(D-3)[g]=0

Leth(x)=sin2x

Then

(D2+1)[h]=0

Hence

(D-3)D2+1[g-h]=0

Then, every solution to the given nonhomogeneous equation also satisfies

.(D-3)D2+1(D+2)(D+4)[y]=0

Then

y(x)=c1e-2x+c2e-4x+c3e3x+c4sinx+c5cosx (2)

is the general solution to this homogeneous equation

We knowu(x)=uh+up

Comparing (1) & (2)

yp(x)=c3e3x+c4sinx+c5cosx

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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.

xy'''-3y'+exy=x2-1y-2=1,鈥�鈥�y'-2=0,鈥�鈥�y''-2=2

use the annihilator method to determinethe form of a particular solution for the given equation. 胃''-胃=xex

Use the annihilator method to determine the form of a particular solution for the given equation.

(a)y''+6y'+5y=e-x+x2-1

(b)y'''+2y''-19y'-20y=xe-x

(c)y(4)+6y''+9y=x2-sin3x

(d)y'''-y''+2y=xsinx

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