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Chapter 6: Q27E (page 337)

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

y''+2y'+2y=e-xcosx+x2

Short Answer

Expert verified

yp(x)=c1+c2x+c3x2+c6xe-xcosx+c7xe-xsinx

Step by step solution

01

Solve the homogeneous of the given equation

The homogeneous of the given equation is

(D2+2D+2)[y]=0

The solution of the homogeneous is

yh(x)=c1e-xcosx+c2e-xsinx (1)

Now e-xcosx+x2is annihilated byD3D2+2D+2

Then, every solution to the given nonhomogeneous equation also satisfies

.D3D2+2D+22[y]=0

Then

y(x)=c1+c2x+c3x2+c4e-xcosx+c5e-xsinx+ c3x2+c6xe-xcosx+c7xe-xsinx (2)

is the general solution to this homogeneous equation

We knowu(x)=uh+up

Comparing (1) & (2)

yp(x)=c1+c2x+c3x2+c6xe-xcosx+c7xe-xsinx

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