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Chapter 6: Q29E (page 337)

use the annihilator method to determinethe form of a particular solution for the given equation. z'''-2z''+z'=x-ex

Short Answer

Expert verified

zp(x)=c3x2ex+c5x+c6x2

Step by step solution

01

Solve the homogeneous of the given equation

The homogeneous of the given equation is

(D3-2D2+D)[z]=0

The solution of the homogeneous is

zh(x)=c1ex+c2xex+c3 (1)

Nowx-ex is annihilated byD3-D2

Then, every solution to the given nonhomogeneous equation also satisfies

.D3-D2(D3-2D2+D)[z]=0

Then

z(x)=c1ex+c2xex+c3x2ex+c4+c5x+c6x2 (2)

is the general solution to this homogeneous equation

We knowu(x)=uh+up

Comparing (1) & (2)

zp(x)=c3x2ex+c5x+c6x2

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