\[ \left \{x''(t)=c^2 x(t) \left (3 \cos ^2(a t+b)-1\right )+\frac {3}{2} c^2 y(t) \sin (2 a b t),y''(t)=\frac {3}{2} c^2 x(t) \sin (2 a b t)+c^2 y(t) \left (3 \sin ^2(a t+b)-1\right )\right \} \] ✗ Mathematica : cpu = 2.05366 (sec), leaf count = 0 , could not solve
DSolve[{Derivative[2][x][t] == c^2*(-1 + 3*Cos[b + a*t]^2)*x[t] + (3*c^2*Sin[2*a*b*t]*y[t])/2, Derivative[2][y][t] == (3*c^2*Sin[2*a*b*t]*x[t])/2 + c^2*(-1 + 3*Sin[b + a*t]^2)*y[t]}, {x[t], y[t]}, t]
✗ Maple : cpu = 0. (sec), leaf count = 0 , result contains DESol
\[\left \{\left \{x \left (t \right ) = \mathit {DESol}\left (\left \{\left (-36 c^{4} \left (\cos ^{2}\left (b \right )\right ) \left (\cos ^{2}\left (a t \right )\right ) \left (\sin ^{2}\left (b \right )\right ) \left (\sin ^{2}\left (a t \right )\right )-18 c^{4} \cos \left (b \right ) \left (\cos ^{3}\left (a t \right )\right ) \left (\sin ^{3}\left (b \right )\right ) \sin \left (a t \right )+18 c^{4} \cos \left (b \right ) \cos \left (a t \right ) \left (\sin ^{3}\left (b \right )\right ) \left (\sin ^{3}\left (a t \right )\right )+c^{4}-3 c^{4} \left (\sin ^{2}\left (b \right )\right ) \left (\sin ^{2}\left (a t \right )\right )-9 c^{4} \left (\cos ^{2}\left (a b t \right )\right ) \left (\sin ^{2}\left (a b t \right )\right )-3 c^{4} \left (\cos ^{2}\left (b \right )\right ) \left (\sin ^{2}\left (a t \right )\right )-3 c^{4} \left (\cos ^{2}\left (a t \right )\right ) \left (\sin ^{2}\left (b \right )\right )-3 c^{4} \left (\cos ^{2}\left (b \right )\right ) \left (\cos ^{2}\left (a t \right )\right )-24 a^{2} c^{2} \cos \left (b \right ) \cos \left (a t \right ) \sin \left (b \right ) \sin \left (a t \right )+6 a^{2} c^{2} \left (\cos ^{2}\left (b \right )\right ) \left (\cos ^{2}\left (a t \right )\right )-6 a^{2} c^{2} \left (\cos ^{2}\left (b \right )\right ) \left (\sin ^{2}\left (a t \right )\right )-6 a^{2} c^{2} \left (\cos ^{2}\left (a t \right )\right ) \left (\sin ^{2}\left (b \right )\right )+6 a^{2} c^{2} \left (\sin ^{2}\left (b \right )\right ) \left (\sin ^{2}\left (a t \right )\right )+9 c^{4} \left (\cos ^{2}\left (a t \right )\right ) \left (\sin ^{4}\left (b \right )\right ) \left (\sin ^{2}\left (a t \right )\right )+9 c^{4} \left (\cos ^{2}\left (b \right )\right ) \left (\cos ^{4}\left (a t \right )\right ) \left (\sin ^{2}\left (b \right )\right )+9 c^{4} \left (\cos ^{2}\left (b \right )\right ) \left (\sin ^{2}\left (b \right )\right ) \left (\sin ^{4}\left (a t \right )\right )+9 c^{4} \left (\cos ^{4}\left (b \right )\right ) \left (\cos ^{2}\left (a t \right )\right ) \left (\sin ^{2}\left (a t \right )\right )+\frac {12 a^{2} b^{2} c^{2} \cos \left (b \right ) \cos \left (a t \right ) \left (\cos ^{2}\left (a b t \right )\right ) \sin \left (b \right ) \sin \left (a t \right )}{\sin \left (a b t \right )^{2}}+\frac {12 a^{2} b^{2} c^{2} \cos \left (b \right ) \cos \left (a t \right ) \sin \left (b \right ) \sin \left (a t \right ) \left (\sin ^{2}\left (a b t \right )\right )}{\cos \left (a b t \right )^{2}}+\frac {2 a^{2} b^{2} c^{2} \left (\cos ^{2}\left (a b t \right )\right )}{\sin \left (a b t \right )^{2}}+\frac {2 a^{2} b^{2} c^{2} \left (\sin ^{2}\left (a b t \right )\right )}{\cos \left (a b t \right )^{2}}-\frac {6 a^{2} b^{2} c^{2} \left (\cos ^{2}\left (b \right )\right ) \left (\cos ^{2}\left (a t \right )\right ) \left (\cos ^{2}\left (a b t \right )\right )}{\sin \left (a b t \right )^{2}}-\frac {12 a^{2} b \,c^{2} \left (\cos ^{2}\left (b \right )\right ) \cos \left (a t \right ) \cos \left (a b t \right ) \sin \left (a t \right )}{\sin \left (a b t \right )}+\frac {12 a^{2} b \,c^{2} \left (\cos ^{2}\left (b \right )\right ) \cos \left (a t \right ) \sin \left (a t \right ) \sin \left (a b t \right )}{\cos \left (a b t \right )}-\frac {12 a^{2} b \,c^{2} \cos \left (b \right ) \left (\cos ^{2}\left (a t \right )\right ) \cos \left (a b t \right ) \sin \left (b \right )}{\sin \left (a b t \right )}+\frac {12 a^{2} b \,c^{2} \cos \left (b \right ) \left (\cos ^{2}\left (a t \right )\right ) \sin \left (b \right ) \sin \left (a b t \right )}{\cos \left (a b t \right )}+\frac {12 a^{2} b \,c^{2} \cos \left (b \right ) \cos \left (a b t \right ) \sin \left (b \right ) \left (\sin ^{2}\left (a t \right )\right )}{\sin \left (a b t \right )}-\frac {12 a^{2} b \,c^{2} \cos \left (b \right ) \sin \left (b \right ) \left (\sin ^{2}\left (a t \right )\right ) \sin \left (a b t \right )}{\cos \left (a b t \right )}+\frac {12 a^{2} b \,c^{2} \cos \left (a t \right ) \cos \left (a b t \right ) \left (\sin ^{2}\left (b \right )\right ) \sin \left (a t \right )}{\sin \left (a b t \right )}-\frac {12 a^{2} b \,c^{2} \cos \left (a t \right ) \left (\sin ^{2}\left (b \right )\right ) \sin \left (a t \right ) \sin \left (a b t \right )}{\cos \left (a b t \right )}+18 c^{4} \left (\cos ^{3}\left (b \right )\right ) \left (\cos ^{3}\left (a t \right )\right ) \sin \left (b \right ) \sin \left (a t \right )-18 c^{4} \left (\cos ^{3}\left (b \right )\right ) \cos \left (a t \right ) \sin \left (b \right ) \left (\sin ^{3}\left (a t \right )\right )-\frac {6 a^{2} b^{2} c^{2} \left (\cos ^{2}\left (b \right )\right ) \left (\cos ^{2}\left (a t \right )\right ) \left (\sin ^{2}\left (a b t \right )\right )}{\cos \left (a b t \right )^{2}}-\frac {6 a^{2} b^{2} c^{2} \left (\cos ^{2}\left (a b t \right )\right ) \left (\sin ^{2}\left (b \right )\right ) \left (\sin ^{2}\left (a t \right )\right )}{\sin \left (a b t \right )^{2}}-\frac {6 a^{2} b^{2} c^{2} \left (\sin ^{2}\left (b \right )\right ) \left (\sin ^{2}\left (a t \right )\right ) \left (\sin ^{2}\left (a b t \right )\right )}{\cos \left (a b t \right )^{2}}\right ) \textit {\_Y} \left (t \right )+\left (\frac {6 a b \,c^{2} \left (\cos ^{2}\left (b \right )\right ) \left (\cos ^{2}\left (a t \right )\right ) \cos \left (a b t \right )}{\sin \left (a b t \right )}-\frac {6 a b \,c^{2} \left (\cos ^{2}\left (b \right )\right ) \left (\cos ^{2}\left (a t \right )\right ) \sin \left (a b t \right )}{\cos \left (a b t \right )}-\frac {12 a b \,c^{2} \cos \left (b \right ) \cos \left (a t \right ) \cos \left (a b t \right ) \sin \left (b \right ) \sin \left (a t \right )}{\sin \left (a b t \right )}+\frac {12 a b \,c^{2} \cos \left (b \right ) \cos \left (a t \right ) \sin \left (b \right ) \sin \left (a t \right ) \sin \left (a b t \right )}{\cos \left (a b t \right )}+\frac {6 a b \,c^{2} \cos \left (a b t \right ) \left (\sin ^{2}\left (b \right )\right ) \left (\sin ^{2}\left (a t \right )\right )}{\sin \left (a b t \right )}-\frac {6 a b \,c^{2} \left (\sin ^{2}\left (b \right )\right ) \left (\sin ^{2}\left (a t \right )\right ) \sin \left (a b t \right )}{\cos \left (a b t \right )}+12 a \,c^{2} \left (\cos ^{2}\left (b \right )\right ) \cos \left (a t \right ) \sin \left (a t \right )+12 a \,c^{2} \cos \left (b \right ) \left (\cos ^{2}\left (a t \right )\right ) \sin \left (b \right )-12 a \,c^{2} \cos \left (b \right ) \sin \left (b \right ) \left (\sin ^{2}\left (a t \right )\right )-12 a \,c^{2} \cos \left (a t \right ) \left (\sin ^{2}\left (b \right )\right ) \sin \left (a t \right )-\frac {2 a b \,c^{2} \cos \left (a b t \right )}{\sin \left (a b t \right )}+\frac {2 a b \,c^{2} \sin \left (a b t \right )}{\cos \left (a b t \right )}\right ) \left (\frac {d}{d t}\textit {\_Y} \left (t \right )\right )+\left (-3 c^{2} \left (\cos ^{2}\left (b \right )\right ) \left (\cos ^{2}\left (a t \right )\right )-3 c^{2} \left (\cos ^{2}\left (b \right )\right ) \left (\sin ^{2}\left (a t \right )\right )-3 c^{2} \left (\cos ^{2}\left (a t \right )\right ) \left (\sin ^{2}\left (b \right )\right )-3 c^{2} \left (\sin ^{2}\left (b \right )\right ) \left (\sin ^{2}\left (a t \right )\right )+\frac {2 a^{2} b^{2} \left (\cos ^{2}\left (a b t \right )\right )}{\sin \left (a b t \right )^{2}}+\frac {2 a^{2} b^{2} \left (\sin ^{2}\left (a b t \right )\right )}{\cos \left (a b t \right )^{2}}+2 c^{2}\right ) \left (\frac {d^{2}}{d t^{2}}\textit {\_Y} \left (t \right )\right )+\left (-\frac {2 a b \cos \left (a b t \right )}{\sin \left (a b t \right )}+\frac {2 a b \sin \left (a b t \right )}{\cos \left (a b t \right )}\right ) \left (\frac {d^{3}}{d t^{3}}\textit {\_Y} \left (t \right )\right )+\frac {d^{4}}{d t^{4}}\textit {\_Y} \left (t \right )\right \}, \left \{\textit {\_Y} \left (t \right )\right \}\right ), y \left (t \right ) = \frac {-3 c^{2} \left (\cos ^{2}\left (b \right )\right ) \left (\cos ^{2}\left (a t \right )\right ) x \left (t \right )+6 c^{2} \cos \left (b \right ) \cos \left (a t \right ) \sin \left (b \right ) \sin \left (a t \right ) x \left (t \right )-3 c^{2} \left (\sin ^{2}\left (b \right )\right ) \left (\sin ^{2}\left (a t \right )\right ) x \left (t \right )+c^{2} x \left (t \right )+\frac {d^{2}}{d t^{2}}x \left (t \right )}{3 c^{2} \cos \left (a b t \right ) \sin \left (a b t \right )}\right \}\right \}\]