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

$ D^3 + 1 = 0 \\ $

$D = -1, \frac {1\pm \sqrt{3}i} {2} \\ $

$ \text{ C.F. is } y_c = c_1e^{-x} + e^\frac{x}{2}[c_2 cos\frac{\sqrt{3}}{2}x + c_3sin\frac{\sqrt{3}}{2}x]\\ $

$ \text{P.I = }y_p = \frac{1}{D^3 + 1}e^\frac{x}{2}sin\frac{\sqrt{3}}{2}x\\ $

$= e^\frac{x}{2}\frac{1}{(D + \frac{1}{2})^3 + 1}sin\frac{\sqrt{3}}{2}x $

$= e^\frac{x}{2}\frac{1}{D^3 + \frac{3}{2}D^2 + \frac{3}{4}D + \frac{9}{8}}sin\frac{\sqrt{3}}{2}x $

$ \text{If we put }D^2 = - \frac{3}{4} \text{, the denominator vanishes.} \\ $

$ \therefore y_p = e^\frac{x}{2}\frac{x}{3D^2 + 3D + \frac{3}{4}}sin\frac{\sqrt{3}}{2}x \\ $

$ = e^\frac{x}{2}\frac{x}{3(\frac{-3}{4}) + 3D + \frac{3}{4}}sin\frac{\sqrt{3}}{2}x \\ $

$ = e^\frac{x}{2}\frac{x}{ 3D - \frac{3}{2}}sin\frac{\sqrt{3}}{2}x \\ $

$ = 2e^\frac{x}{2}x\frac{6D+3}{ 36D^2 - 9}sin\frac{\sqrt{3}}{2}x \\ $

$ = 2e^\frac{x}{2}x\frac{6D+3}{ 36(\frac{-3}{4}) - 9}sin\frac{\sqrt{3}}{2}x \\ $

$ = 2e^\frac{x}{2}x\frac{6D+3}{ -36}sin\frac{\sqrt{3}}{2}x \\ $

$ = \frac{-2e^\frac{x}{2}}{36}x[6 \frac{\sqrt3}{2}cos\frac{\sqrt{3}}{2}x + 3 sin\frac{\sqrt{3}}{2}x ] \\ $

$ = -x\frac{e^\frac{x}{2}}{6}[\sqrt3cos\frac{\sqrt{3}}{2}x + sin\frac{\sqrt{3}}{2}x ] \\ $

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

$ \therefore y = c_1e^{-x} + e^\frac{x}{2}[c_2 cos\frac{\sqrt{3}}{2}x + c_3sin\frac{\sqrt{3}}{2}x] -x\frac{e^\frac{x}{2}}{6}[\sqrt3cos\frac{\sqrt{3}}{2}x + sin\frac{\sqrt{3}}{2}x ] \\ $