$ \text{ The Auxiliary equation is } \\ $

$ D^2 + 2D + 1 = 0 \\ $

$D = -1, -1 \\ $

$ \text{ C.F. is } y_c = (c_1 + c_2x )e^{-x}\\ $

$ \text{P.I = }y_p = \frac{1}{D^2 + 2D + 1}xe^{-x}cosx\\ $

$= \frac{1}{(D+1)^2}xe^{-x}cosx \\ $

$=e^{-x} \frac{1}{(D-1 +1)^2}xcosx \\ $

$=e^{-x} \frac{1}{D^2}xcosx \\ $

$=e^{-x} \frac{1}{D}\int xcosx dx \\ $

$=e^{-x} \frac{1}{D}[ xsinx + cosx ] \\ $

$=e^{-x} \int [ xsinx + cosx ] dx \\ $

$=e^{-x} [ x(-cosx) - (-sinx )+ sinx ] \\ $

$=e^{-x} [ -xcosx + 2sinx ] \\ $

$\therefore \text{ The complete solution is y = C.F. + P.I. } \\ $

$ \therefore y = (c_1 + c_2x )e^{-x} + e^{-x} [ -xcosx + 2sinx ] \\ $