Use the method of undetermined coefficients to solve the equation: y'' + 2y' + 2y = e^(-x)cos(x)?

Since D² - 2αD + ( α² + β² ) annihilates e^(αx)*cos βx ,
take α = - 1 , β = 1 , we get
( D² + 2D + 2 )[ e^(-x)*cos x ] = 0
( D² + 2D + 2 )( y'' + 2y' + 2y ) = 0
( D² + 2D + 2 )( D² + 2D + 2 )y = 0
( D² + 2D + 2 )² y = 0
( 1 + i ) is a complex root of multiplicity 2 ,
and also ( 1 - i ) is a complex root of multiplicity 2 .
y = e^x( c1*cos x + c2*sin x ) + x*e^x( c3*cos x + c4*sin x ) ..... Ans