\[ y(x) \left (a^4 \sin ^4(x)-3\right )+y^{(4)}(x) \sin ^4(x)+2 y^{(3)}(x) \sin ^3(x) \cos (x)+\left (\sin ^2(x)-3\right ) \sin ^2(x) y''(x)+\left (2 \sin ^2(x)+3\right ) \sin (x) \cos (x) y'(x)=0 \] ✓ Mathematica : cpu = 0.161204 (sec), leaf count = 270
DSolve[(-3 + a^4*Sin[x]^4)*y[x] + Cos[x]*Sin[x]*(3 + 2*Sin[x]^2)*Derivative[1][y][x] + Sin[x]^2*(-3 + Sin[x]^2)*Derivative[2][y][x] + 2*Cos[x]*Sin[x]^3*Derivative[3][y][x] + Sin[x]^4*Derivative[4][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \sin (x) \, _2F_1\left (\frac {1}{4} \left (3-\sqrt {5-4 \sqrt {1-a^4}}\right ),\frac {1}{4} \left (\sqrt {5-4 \sqrt {1-a^4}}+3\right );\frac {1}{2};\cos ^2(x)\right )+c_3 \sin (x) \cos (x) \, _2F_1\left (\frac {1}{4} \left (5-\sqrt {5-4 \sqrt {1-a^4}}\right ),\frac {1}{4} \left (\sqrt {5-4 \sqrt {1-a^4}}+5\right );\frac {3}{2};\cos ^2(x)\right )+c_2 \sin (x) \, _2F_1\left (\frac {1}{4} \left (3-\sqrt {4 \sqrt {1-a^4}+5}\right ),\frac {1}{4} \left (\sqrt {4 \sqrt {1-a^4}+5}+3\right );\frac {1}{2};\cos ^2(x)\right )+c_4 \sin (x) \cos (x) \, _2F_1\left (\frac {1}{4} \left (5-\sqrt {4 \sqrt {1-a^4}+5}\right ),\frac {1}{4} \left (\sqrt {4 \sqrt {1-a^4}+5}+5\right );\frac {3}{2};\cos ^2(x)\right )\right \}\right \}\] ✓ Maple : cpu = 0.67 (sec), leaf count = 252
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)*sin(x)^4+2*diff(diff(diff(y(x),x),x),x)*sin(x)^3*cos(x)+diff(diff(y(x),x),x)*sin(x)^2*(sin(x)^2-3)+diff(y(x),x)*sin(x)*cos(x)*(2*sin(x)^2+3)+(a^4*sin(x)^4-3)*y(x)=0,y(x))
\[y \left (x \right ) = \sin \left (x \right ) \left (\hypergeom \left (\left [\frac {3}{4}+\frac {\sqrt {-4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}, \frac {3}{4}-\frac {\sqrt {-4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}\right ], \left [\frac {1}{2}\right ], \cos ^{2}\left (x \right )\right ) c_{1}+\hypergeom \left (\left [\frac {3}{4}+\frac {\sqrt {4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}, \frac {3}{4}-\frac {\sqrt {4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}\right ], \left [\frac {1}{2}\right ], \cos ^{2}\left (x \right )\right ) c_{2}+\cos \left (x \right ) \left (\hypergeom \left (\left [\frac {5}{4}+\frac {\sqrt {-4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}, \frac {5}{4}-\frac {\sqrt {-4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}\right ], \left [\frac {3}{2}\right ], \cos ^{2}\left (x \right )\right ) c_{3}+\hypergeom \left (\left [\frac {5}{4}-\frac {\sqrt {4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}, \frac {5}{4}+\frac {\sqrt {4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}\right ], \left [\frac {3}{2}\right ], \cos ^{2}\left (x \right )\right ) c_{4}\right )\right )\]