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

$ D^2 + 1 = 0 \\ $

$D = \pm i \\ $

$ \text{ C.F. is } y_c = c_1cosx + c_2sinx\\ $

$ \text{P.I = }y_p = \frac{1}{D^2 + 1}cosecx\\ $

$=\frac{1}{(D+i)(D-i)}cosecx\\ $

$=\frac{-1}{2i} [\frac{1}{D+i} - \frac{1}{D-i}] cosecx\\ $

$=\frac{1}{2i} [\frac{1}{D-i} - \frac{1}{D+i}] cosecx\\ $

$\text {Consider} \frac{1}{(D+i) }cosecx \\ $

$ = e^{-ix} \int e^{ix}cosecx dx \\ $

$ = e^{-ix} \int (cosx + i sinx )cosecx dx \\ $

$ = e^{-ix} \int (cotx + i ) dx \\ $

$ = e^{-ix} [log sinx + ix] \\ $

$ \frac{1}{(D-i) }cosecx = e^{+ix} [log sinx - ix] \\ $

$ \text{P.I = } \frac{1}{2i} [ e^{ix} (logsinx - ix) - e^{-ix} ( logsinx + ix ) ] \\ $

$ = log sinx \frac{(e^{ix} - e^{-ix})}{2i} - x \frac{(e^{ix} + e^{-ix})} {2} \\ $

$ = (log sinx) sinx - x cosx \\ $

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

$ \therefore y = c_1cosx + c_2sinx + (log sinx) sinx - x cosx \\ $