\[ y^{(6)}(x)+y(x)-\sin \left (\frac {x}{2}\right ) \sin \left (\frac {3 x}{2}\right )=0 \] ✓ Mathematica : cpu = 2.38973 (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 = 0.659 (sec), leaf count = 79
\[ \left \{ y \left ( x \right ) ={\frac {1}{504} \left ( 504\,\cos \left ( x/2 \right ) {\it \_C3}+504\,\sin \left ( x/2 \right ) {\it \_C4} \right ) {{\rm e}^{-{\frac {\sqrt {3}x}{2}}}}}+{\frac {1}{504} \left ( 504\,\cos \left ( x/2 \right ) {\it \_C5}+504\,\sin \left ( x/2 \right ) {\it \_C6} \right ) {{\rm e}^{{\frac {\sqrt {3}x}{2}}}}}+{\frac {\cos \left ( 2\,x \right ) }{126}}+{\frac { \left ( 504\,{\it \_C1}+105 \right ) \cos \left ( x \right ) }{504}}+{\frac { \left ( 42\,x+504\,{\it \_C2} \right ) \sin \left ( x \right ) }{504}} \right \} \]