\[ y^{(6)}(x)+y(x)-\sin \left (\frac {x}{2}\right ) \sin \left (\frac {3 x}{2}\right )=0 \] ✓ Mathematica : cpu = 1.19336 (sec), leaf count = 111
\[\left \{\left \{y(x)\to c_4 e^{-\frac {\sqrt {3} x}{2}} \sin \left (\frac {x}{2}\right )+c_6 e^{\frac {\sqrt {3} x}{2}} \sin \left (\frac {x}{2}\right )+c_5 \sin (x)+e^{-\frac {\sqrt {3} x}{2}} \left (c_1 e^{\sqrt {3} x}+c_3\right ) \cos \left (\frac {x}{2}\right )+\left (c_2+\frac {1}{4}\right ) \cos (x)+\frac {1}{12} x \sin (x)+\frac {1}{126} \cos (2 x)\right \}\right \}\]
✓ Maple : cpu = 0.841 (sec), leaf count = 147
\[ \left \{ y \left ( x \right ) ={\frac {1}{504} \left ( 504\,{\it \_C3}\,\cos \left ( x/2 \right ) +504\,{\it \_C4}\,\sin \left ( x/2 \right ) \right ) {{\rm e}^{-{\frac {\sqrt {3}x}{2}}}}}+{\frac {1}{504} \left ( 504\,{\it \_C5}\,\cos \left ( x/2 \right ) +504\,{\it \_C6}\,\sin \left ( x/2 \right ) \right ) {{\rm e}^{{\frac {\sqrt {3}x}{2}}}}}+{\frac {{{\rm e}^{-{\frac {i}{2}}x}}}{504} \left ( -21\,i\sin \left ( {\frac {x}{2}} \right ) +33\,i\sin \left ( {\frac {3\,x}{2}} \right ) +21\,\cos \left ( x/2 \right ) +9\,\cos \left ( 3/2\,x \right ) \right ) }+{\frac {{{\rm e}^{{\frac {i}{2}}x}}}{504} \left ( 21\,i\sin \left ( {\frac {x}{2}} \right ) -33\,i\sin \left ( {\frac {3\,x}{2}} \right ) +21\,\cos \left ( x/2 \right ) +9\,\cos \left ( 3/2\,x \right ) \right ) }+{\frac {\cos \left ( 2\,x \right ) }{18}}+{\frac { \left ( 504\,{\it \_C1}+21 \right ) \cos \left ( x \right ) }{504}}+{\frac { \left ( 42\,x+504\,{\it \_C2} \right ) \sin \left ( x \right ) }{504}} \right \} \]