\[ y^{(6)}(x)+y(x)-\sin \left (\frac {x}{2}\right ) \sin \left (\frac {3 x}{2}\right )=0 \] ✓ Mathematica : cpu = 0.968032 (sec), leaf count = 234
\[\left \{\left \{y(x)\to c_4 e^{-\frac {\sqrt {3} x}{2}} \sin \left (\frac {x}{2}\right )+c_6 e^{\frac {\sqrt {3} x}{2}} \sin \left (\frac {x}{2}\right )+c_5 \sin (x)+c_1 e^{\frac {\sqrt {3} x}{2}} \cos \left (\frac {x}{2}\right )+c_3 e^{-\frac {\sqrt {3} x}{2}} \cos \left (\frac {x}{2}\right )+c_2 \cos (x)+\frac {1}{504} \left (-42 \sin ^2\left (\frac {x}{2}\right )-42 \sin ^2(x)+42 x \sin (x)+42 \sin \left (\frac {x}{2}\right ) \sin \left (\frac {3 x}{2}\right )+21 \sin (x) \sin (2 x)-24 \sin \left (\frac {x}{2}\right ) \sin \left (\frac {5 x}{2}\right )-14 \sin (x) \sin (3 x)-28 \cos ^4(x)+42 \cos ^3(x)+63 \cos ^2(x)+42 \cos ^2\left (\frac {x}{2}\right )-7 \cos (3 x) \cos (x)+42 \cos \left (\frac {x}{2}\right ) \cos \left (\frac {3 x}{2}\right )-24 \cos \left (\frac {x}{2}\right ) \cos \left (\frac {5 x}{2}\right )\right )\right \}\right \}\]
✓ Maple : cpu = 0.805 (sec), leaf count = 164
\[ \left \{ y \left ( x \right ) = \left ( {\frac {{{\rm e}^{{\frac {i}{2}}x}}}{24}}+{\frac {{{\rm e}^{-{\frac {i}{2}}x}}}{24}} \right ) \cos \left ( {\frac {x}{2}} \right ) + \left ( {\frac {{{\rm e}^{{\frac {i}{2}}x}}}{56}}+{\frac {{{\rm e}^{-{\frac {i}{2}}x}}}{56}} \right ) \cos \left ( {\frac {3\,x}{2}} \right ) + \left ( {\frac {i}{24}}{{\rm e}^{{\frac {i}{2}}x}}-{\frac {i}{24}}{{\rm e}^{-{\frac {i}{2}}x}} \right ) \sin \left ( {\frac {x}{2}} \right ) + \left ( -{\frac {11\,i}{168}}{{\rm e}^{{\frac {i}{2}}x}}+{\frac {11\,i}{168}}{{\rm e}^{-{\frac {i}{2}}x}} \right ) \sin \left ( {\frac {3\,x}{2}} \right ) +{\frac {x\sin \left ( x \right ) }{12}}+{\frac {\cos \left ( x \right ) }{24}}+{\frac {\cos \left ( 2\,x \right ) }{18}}+{\it \_C1}\,\cos \left ( x \right ) +{\it \_C2}\,\sin \left ( x \right ) +{\it \_C3}\,{{\rm e}^{-{\frac {\sqrt {3}x}{2}}}}\cos \left ( {\frac {x}{2}} \right ) +{\it \_C4}\,{{\rm e}^{-{\frac {\sqrt {3}x}{2}}}}\sin \left ( {\frac {x}{2}} \right ) +{\it \_C5}\,{{\rm e}^{{\frac {\sqrt {3}x}{2}}}}\cos \left ( {\frac {x}{2}} \right ) +{\it \_C6}\,{{\rm e}^{{\frac {\sqrt {3}x}{2}}}}\sin \left ( {\frac {x}{2}} \right ) \right \} \]