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

$ D^4 + 8D^2 +16 = 0 \\ $

$ (D^2 + 4)^2 = 0 \\ $

$D=\pm 2i, \pm2i \\ $

$ \text{ C.F. is } y = (c_1 + c_2x)cos2x + (c_3 +c_4x)sin2x\\ $

$ \text{P.I = } \frac{1}{D^4 + 8D^2 +16}sin^2x \\ $

$= \frac{1}{D^4 + 8D^2 +16} \frac{(1-cos2x)}{2} $

$= \frac{1}{2}[ \frac{1}{D^4 + 8D^2 +16}e^{0x} - \frac{1}{D^4 + 8D^2 +16} cos2x ]\\ $

$= \frac{1}{2}[ \frac{1}{16} - \frac{x}{4D^3 + 16D} cos2x ]\\ $

$= \frac{1}{2}[ \frac{1}{16} - \frac{x^2}{12D^2 + 16} cos2x ]\\ $

$= \frac{1}{2}[ \frac{1}{16} - \frac{x^2}{12(-4) + 16} cos2x ]\\ $

$= \frac{1}{2}[ \frac{1}{16} + \frac{x^2}{32} cos2x ]\\ $

$= \frac{1}{32} + \frac{x^2}{64} cos2x \\ $

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

$ \therefore y = (c_1 + c_2x)cos2x + (c_3 +c_4x)sin2x+ \frac{1}{32} + \frac{x^2}{64} cos2x \\ $