\[ \left \{x''(t)=c^2 x(t) \left (3 \cos ^2(a t+b)-1\right )+\frac {3}{2} c^2 y(t) \sin (2 a b t),y''(t)=\frac {3}{2} c^2 x(t) \sin (2 a b t)+c^2 y(t) \left (3 \sin ^2(a t+b)-1\right )\right \} \] ✗ Mathematica : cpu = 1.69863 (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 or ODESolStruc
\[\left \{x \relax (t ) = \mathit {DESol}\left (\left \{\frac {d^{4}}{d t^{4}}\textit {\_Y} \relax (t )+\left (\frac {2 \sin \left (a t b \right ) a b}{\cos \left (a t b \right )}-\frac {2 \cos \left (a t b \right ) a b}{\sin \left (a t b \right )}\right ) \left (\frac {d^{3}}{d t^{3}}\textit {\_Y} \relax (t )\right )+\left (\frac {2 \left (\sin ^{2}\left (a t b \right )\right ) a^{2} b^{2}}{\cos \left (a t b \right )^{2}}+\frac {2 \left (\cos ^{2}\left (a t b \right )\right ) a^{2} b^{2}}{\sin \left (a t b \right )^{2}}+2 c^{2}-3 \left (\sin ^{2}\left (a t \right )\right ) c^{2} \left (\cos ^{2}\relax (b )\right )-3 \left (\cos ^{2}\left (a t \right )\right ) c^{2} \left (\cos ^{2}\relax (b )\right )-3 \left (\sin ^{2}\left (a t \right )\right ) c^{2} \left (\sin ^{2}\relax (b )\right )-3 \left (\cos ^{2}\left (a t \right )\right ) c^{2} \left (\sin ^{2}\relax (b )\right )\right ) \left (\frac {d^{2}}{d t^{2}}\textit {\_Y} \relax (t )\right )+\left (12 \left (\cos ^{2}\left (a t \right )\right ) a \,c^{2} \cos \relax (b ) \sin \relax (b )-12 \sin \left (a t \right ) \cos \left (a t \right ) a \,c^{2} \left (\sin ^{2}\relax (b )\right )+\frac {2 \sin \left (a t b \right ) a b \,c^{2}}{\cos \left (a t b \right )}-\frac {2 \cos \left (a t b \right ) a b \,c^{2}}{\sin \left (a t b \right )}+12 \sin \left (a t \right ) \cos \left (a t \right ) a \,c^{2} \left (\cos ^{2}\relax (b )\right )-12 \left (\sin ^{2}\left (a t \right )\right ) a \,c^{2} \cos \relax (b ) \sin \relax (b )-\frac {6 \sin \left (a t b \right ) \left (\cos ^{2}\left (a t \right )\right ) a b \,c^{2} \left (\cos ^{2}\relax (b )\right )}{\cos \left (a t b \right )}+\frac {6 \cos \left (a t b \right ) \left (\cos ^{2}\left (a t \right )\right ) a b \,c^{2} \left (\cos ^{2}\relax (b )\right )}{\sin \left (a t b \right )}-\frac {6 \sin \left (a t b \right ) \left (\sin ^{2}\left (a t \right )\right ) a b \,c^{2} \left (\sin ^{2}\relax (b )\right )}{\cos \left (a t b \right )}+\frac {6 \cos \left (a t b \right ) \left (\sin ^{2}\left (a t \right )\right ) a b \,c^{2} \left (\sin ^{2}\relax (b )\right )}{\sin \left (a t b \right )}+\frac {12 \sin \left (a t b \right ) \sin \left (a t \right ) \cos \left (a t \right ) a b \,c^{2} \cos \relax (b ) \sin \relax (b )}{\cos \left (a t b \right )}-\frac {12 \cos \left (a t b \right ) \sin \left (a t \right ) \cos \left (a t \right ) a b \,c^{2} \cos \relax (b ) \sin \relax (b )}{\sin \left (a t b \right )}\right ) \left (\frac {d}{d t}\textit {\_Y} \relax (t )\right )+\left (c^{4}-\frac {6 \left (\cos ^{2}\left (a t b \right )\right ) \left (\cos ^{2}\left (a t \right )\right ) a^{2} b^{2} c^{2} \left (\cos ^{2}\relax (b )\right )}{\sin \left (a t b \right )^{2}}-\frac {6 \left (\sin ^{2}\left (a t b \right )\right ) \left (\sin ^{2}\left (a t \right )\right ) a^{2} b^{2} c^{2} \left (\sin ^{2}\relax (b )\right )}{\cos \left (a t b \right )^{2}}-\frac {6 \left (\cos ^{2}\left (a t b \right )\right ) \left (\sin ^{2}\left (a t \right )\right ) a^{2} b^{2} c^{2} \left (\sin ^{2}\relax (b )\right )}{\sin \left (a t b \right )^{2}}-\frac {6 \left (\sin ^{2}\left (a t b \right )\right ) \left (\cos ^{2}\left (a t \right )\right ) a^{2} b^{2} c^{2} \left (\cos ^{2}\relax (b )\right )}{\cos \left (a t b \right )^{2}}-3 \left (\cos ^{2}\left (a t \right )\right ) c^{4} \left (\sin ^{2}\relax (b )\right )-3 \left (\sin ^{2}\left (a t \right )\right ) c^{4} \left (\sin ^{2}\relax (b )\right )-3 \left (\sin ^{2}\left (a t \right )\right ) c^{4} \left (\cos ^{2}\relax (b )\right )-3 \left (\cos ^{2}\left (a t \right )\right ) c^{4} \left (\cos ^{2}\relax (b )\right )-9 \left (\sin ^{2}\left (a t b \right )\right ) \left (\cos ^{2}\left (a t b \right )\right ) c^{4}+18 \left (\sin ^{3}\left (a t \right )\right ) \cos \left (a t \right ) c^{4} \cos \relax (b ) \left (\sin ^{3}\relax (b )\right )-18 \sin \left (a t \right ) \left (\cos ^{3}\left (a t \right )\right ) c^{4} \cos \relax (b ) \left (\sin ^{3}\relax (b )\right )+\frac {2 \left (\sin ^{2}\left (a t b \right )\right ) a^{2} b^{2} c^{2}}{\cos \left (a t b \right )^{2}}+\frac {2 \left (\cos ^{2}\left (a t b \right )\right ) a^{2} b^{2} c^{2}}{\sin \left (a t b \right )^{2}}-18 \left (\sin ^{3}\left (a t \right )\right ) \cos \left (a t \right ) c^{4} \left (\cos ^{3}\relax (b )\right ) \sin \relax (b )-\frac {12 \cos \left (a t b \right ) \sin \left (a t \right ) \cos \left (a t \right ) a^{2} b \,c^{2} \left (\cos ^{2}\relax (b )\right )}{\sin \left (a t b \right )}-\frac {12 \sin \left (a t b \right ) \left (\sin ^{2}\left (a t \right )\right ) a^{2} b \,c^{2} \cos \relax (b ) \sin \relax (b )}{\cos \left (a t b \right )}+\frac {12 \left (\cos ^{2}\left (a t b \right )\right ) \sin \left (a t \right ) \cos \left (a t \right ) a^{2} b^{2} c^{2} \cos \relax (b ) \sin \relax (b )}{\sin \left (a t b \right )^{2}}+\frac {12 \left (\sin ^{2}\left (a t b \right )\right ) \sin \left (a t \right ) \cos \left (a t \right ) a^{2} b^{2} c^{2} \cos \relax (b ) \sin \relax (b )}{\cos \left (a t b \right )^{2}}-24 \sin \left (a t \right ) \cos \left (a t \right ) a^{2} c^{2} \cos \relax (b ) \sin \relax (b )+18 \sin \left (a t \right ) \left (\cos ^{3}\left (a t \right )\right ) c^{4} \left (\cos ^{3}\relax (b )\right ) \sin \relax (b )-36 \left (\sin ^{2}\left (a t \right )\right ) \left (\cos ^{2}\left (a t \right )\right ) c^{4} \left (\cos ^{2}\relax (b )\right ) \left (\sin ^{2}\relax (b )\right )+9 \left (\sin ^{4}\left (a t \right )\right ) c^{4} \left (\cos ^{2}\relax (b )\right ) \left (\sin ^{2}\relax (b )\right )+9 \left (\cos ^{4}\left (a t \right )\right ) c^{4} \left (\cos ^{2}\relax (b )\right ) \left (\sin ^{2}\relax (b )\right )+9 \left (\sin ^{2}\left (a t \right )\right ) \left (\cos ^{2}\left (a t \right )\right ) c^{4} \left (\sin ^{4}\relax (b )\right )-6 \left (\sin ^{2}\left (a t \right )\right ) a^{2} c^{2} \left (\cos ^{2}\relax (b )\right )+6 \left (\cos ^{2}\left (a t \right )\right ) a^{2} c^{2} \left (\cos ^{2}\relax (b )\right )+6 \left (\sin ^{2}\left (a t \right )\right ) a^{2} c^{2} \left (\sin ^{2}\relax (b )\right )-6 \left (\cos ^{2}\left (a t \right )\right ) a^{2} c^{2} \left (\sin ^{2}\relax (b )\right )+9 \left (\sin ^{2}\left (a t \right )\right ) \left (\cos ^{2}\left (a t \right )\right ) c^{4} \left (\cos ^{4}\relax (b )\right )+\frac {12 \cos \left (a t b \right ) \left (\sin ^{2}\left (a t \right )\right ) a^{2} b \,c^{2} \cos \relax (b ) \sin \relax (b )}{\sin \left (a t b \right )}+\frac {12 \sin \left (a t b \right ) \left (\cos ^{2}\left (a t \right )\right ) a^{2} b \,c^{2} \cos \relax (b ) \sin \relax (b )}{\cos \left (a t b \right )}-\frac {12 \cos \left (a t b \right ) \left (\cos ^{2}\left (a t \right )\right ) a^{2} b \,c^{2} \cos \relax (b ) \sin \relax (b )}{\sin \left (a t b \right )}-\frac {12 \sin \left (a t b \right ) \sin \left (a t \right ) \cos \left (a t \right ) a^{2} b \,c^{2} \left (\sin ^{2}\relax (b )\right )}{\cos \left (a t b \right )}+\frac {12 \cos \left (a t b \right ) \sin \left (a t \right ) \cos \left (a t \right ) a^{2} b \,c^{2} \left (\sin ^{2}\relax (b )\right )}{\sin \left (a t b \right )}+\frac {12 \sin \left (a t b \right ) \sin \left (a t \right ) \cos \left (a t \right ) a^{2} b \,c^{2} \left (\cos ^{2}\relax (b )\right )}{\cos \left (a t b \right )}\right ) \textit {\_Y} \relax (t )\right \}, \left \{\textit {\_Y} \relax (t )\right \}\right ), y \relax (t ) = -\frac {3 x \relax (t ) \left (\sin ^{2}\left (a t \right )\right ) c^{2} \left (\sin ^{2}\relax (b )\right )-6 x \relax (t ) \sin \left (a t \right ) \cos \left (a t \right ) c^{2} \cos \relax (b ) \sin \relax (b )+3 x \relax (t ) \left (\cos ^{2}\left (a t \right )\right ) c^{2} \left (\cos ^{2}\relax (b )\right )-c^{2} x \relax (t )-\frac {d^{2}}{d t^{2}}x \relax (t )}{3 \cos \left (a t b \right ) c^{2} \sin \left (a t b \right )}\right \}\]