\[ y^{(6)}(x)+y(x)-\sin \left (\frac {x}{2}\right ) \sin \left (\frac {3 x}{2}\right )=0 \] ✓ Mathematica : cpu = 3.92771 (sec), leaf count = 234
DSolve[-(Sin[x/2]*Sin[(3*x)/2]) + y[x] + Derivative[6][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {1}{504} \left (-42 \sin ^2\left (\frac {x}{2}\right )-42 \sin ^2(x)+42 x \sin (x)+42 \sin \left (\frac {x}{2}\right ) \sin \left (\frac {3 x}{2}\right )+21 \sin (x) \sin (2 x)-24 \sin \left (\frac {x}{2}\right ) \sin \left (\frac {5 x}{2}\right )-14 \sin (x) \sin (3 x)-28 \cos ^4(x)+42 \cos ^3(x)+63 \cos ^2(x)+42 \cos ^2\left (\frac {x}{2}\right )-7 \cos (3 x) \cos (x)+42 \cos \left (\frac {x}{2}\right ) \cos \left (\frac {3 x}{2}\right )-24 \cos \left (\frac {x}{2}\right ) \cos \left (\frac {5 x}{2}\right )\right )+c_1 e^{\frac {\sqrt {3} x}{2}} \cos \left (\frac {x}{2}\right )+c_3 e^{-\frac {\sqrt {3} x}{2}} \cos \left (\frac {x}{2}\right )+c_2 \cos (x)+c_4 e^{-\frac {\sqrt {3} x}{2}} \sin \left (\frac {x}{2}\right )+c_6 e^{\frac {\sqrt {3} x}{2}} \sin \left (\frac {x}{2}\right )+c_5 \sin (x)\right \}\right \}\] ✓ Maple : cpu = 2.691 (sec), leaf count = 71
dsolve(diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x)+y(x)-sin(3/2*x)*sin(1/2*x)=0,y(x))
\[y \left (x \right ) = \left (\cos \left (\frac {x}{2}\right ) c_{3}+\sin \left (\frac {x}{2}\right ) c_{4}\right ) {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}}+\left (\cos \left (\frac {x}{2}\right ) c_{5}+\sin \left (\frac {x}{2}\right ) c_{6}\right ) {\mathrm e}^{\frac {\sqrt {3}\, x}{2}}+\frac {\cos \left (2 x \right )}{126}+\frac {\left (5+24 c_{1}\right ) \cos \left (x \right )}{24}+\frac {\left (x +12 c_{2}\right ) \sin \left (x \right )}{12}\]