\[ y^{(6)}(x)+y(x)-\sin \left (\frac {x}{2}\right ) \sin \left (\frac {3 x}{2}\right )=0 \] ✓ Mathematica : cpu = 2.12085 (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 = 0.612 (sec), leaf count = 154
\[ \left \{ y \left ( x \right ) ={\frac {1}{ \left ( i\sqrt {3}+3 \right ) \left ( i\sqrt {3}-3 \right ) \left ( i\sqrt {3}+9 \right ) \left ( i\sqrt {3}-9 \right ) } \left ( \left ( 1008\,\cos \left ( x/2 \right ) {\it \_C3}+1008\,\sin \left ( x/2 \right ) {\it \_C4} \right ) {{\rm e}^{-{\frac {\sqrt {3}x}{2}}}}+ \left ( 1008\,\cos \left ( x/2 \right ) {\it \_C5}+1008\,\sin \left ( x/2 \right ) {\it \_C6} \right ) {{\rm e}^{{\frac {\sqrt {3}x}{2}}}}+ \left ( 42\,\cos \left ( x \right ) -48 \right ) \cos \left ( 2\,x \right ) -28\,\cos \left ( x \right ) \cos \left ( 3\,x \right ) +42\,\sin \left ( x \right ) \sin \left ( 2\,x \right ) -28\,\sin \left ( 3\,x \right ) \sin \left ( x \right ) +84\, \left ( \cos \left ( x \right ) \right ) ^{2}+ \left ( 1008\,{\it \_C1}+168 \right ) \cos \left ( x \right ) +84\,\sin \left ( x \right ) \left ( x-\sin \left ( x \right ) +12\,{\it \_C2} \right ) \right ) } \right \} \]