\[ y^{(6)}(x)+y(x)-\sin \left (\frac {x}{2}\right ) \sin \left (\frac {3 x}{2}\right )=0 \] ✓ Mathematica : cpu = 2.44291 (sec), leaf count = 234
\[\left \{\left \{y(x)\to \frac {1}{504} \left (-42 \sin ^2\left (\frac {x}{2}\right )-42 \sin ^2(x)+42 x \sin (x)+42 \sin \left (\frac {x}{2}\right ) \sin \left (\frac {3 x}{2}\right )+21 \sin (x) \sin (2 x)-24 \sin \left (\frac {x}{2}\right ) \sin \left (\frac {5 x}{2}\right )-14 \sin (x) \sin (3 x)-28 \cos ^4(x)+42 \cos ^3(x)+63 \cos ^2(x)+42 \cos ^2\left (\frac {x}{2}\right )-7 \cos (3 x) \cos (x)+42 \cos \left (\frac {x}{2}\right ) \cos \left (\frac {3 x}{2}\right )-24 \cos \left (\frac {x}{2}\right ) \cos \left (\frac {5 x}{2}\right )\right )+c_1 e^{\frac {\sqrt {3} x}{2}} \cos \left (\frac {x}{2}\right )+c_3 e^{-\frac {\sqrt {3} x}{2}} \cos \left (\frac {x}{2}\right )+c_2 \cos (x)+c_4 e^{-\frac {\sqrt {3} x}{2}} \sin \left (\frac {x}{2}\right )+c_6 e^{\frac {\sqrt {3} x}{2}} \sin \left (\frac {x}{2}\right )+c_5 \sin (x)\right \}\right \}\] ✓ Maple : cpu = 1.247 (sec), leaf count = 79
\[\left \{y \left (x \right ) = \frac {\left (504 c_{1}+105\right ) \cos \left (x \right )}{504}+\frac {\cos \left (2 x \right )}{126}+\frac {\left (504 c_{3} \cos \left (\frac {x}{2}\right )+504 c_{4} \sin \left (\frac {x}{2}\right )\right ) {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}}}{504}+\frac {\left (504 c_{5} \cos \left (\frac {x}{2}\right )+504 c_{6} \sin \left (\frac {x}{2}\right )\right ) {\mathrm e}^{\frac {\sqrt {3}\, x}{2}}}{504}+\frac {\left (504 c_{2}+42 x \right ) \sin \left (x \right )}{504}\right \}\]