\[ \boxed { \left \{ {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) +6\,x \left ( t \right ) +7\,y \left ( t \right ) =0,{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) +3\,x \left ( t \right ) +2\,y \left ( t \right ) =2\,t \right \} } \]
Mathematica: cpu = 0.445557 (sec), leaf count = 766 \[ \left \{\left \{x(t)\to -\frac {7}{60} c_4 e^{-t} \left (3 e^{2 t}-2 e^t \sin (3 t)-3\right )+\frac {1}{60} c_2 e^{-t} \left (9 e^{2 t}+14 e^t \sin (3 t)-9\right )-\frac {7}{20} c_3 e^{-t} \left (e^{2 t}-2 e^t \cos (3 t)+1\right )+\frac {1}{20} c_1 e^{-t} \left (3 e^{2 t}+14 e^t \cos (3 t)+3\right )-\frac {7}{200} e^{-t} \left (e^{2 t}-2 e^t \cos (3 t)+1\right ) \left (-\frac {7}{3} e^t \left (3 e^{-2 t} t+3 t+3 e^{-2 t}-3\right )-\frac {2}{9} \sin (3 t)+\frac {2}{3} t \cos (3 t)\right )+\frac {7}{600} e^{-t} \left (3 e^{2 t}+14 e^t \cos (3 t)+3\right ) \left (3 e^{-t} \left (e^{2 t} (t-1)+t+1\right )-\frac {2}{9} \sin (3 t)+\frac {2}{3} t \cos (3 t)\right )-\frac {7}{600} e^{-t} \left (9 e^{2 t}+14 e^t \sin (3 t)-9\right ) \left (e^{-t} \left (e^{2 t} (t-1)-t-1\right )-\frac {2}{3} t \sin (3 t)-\frac {2}{9} \cos (3 t)\right )-\frac {7}{600} e^{-t} \left (3 e^{2 t}-2 e^t \sin (3 t)-3\right ) \left (e^{-t} \left (7 e^{2 t} (t-1)-7 (t+1)\right )+2 t \sin (3 t)+\frac {2}{3} \cos (3 t)\right ),y(t)\to -\frac {1}{20} c_2 e^{-t} \left (3 e^{2 t}-2 e^t \sin (3 t)-3\right )+\frac {1}{20} c_4 e^{-t} \left (7 e^{2 t}+2 e^t \sin (3 t)-7\right )-\frac {3}{20} c_1 e^{-t} \left (e^{2 t}-2 e^t \cos (3 t)+1\right )+\frac {1}{20} c_3 e^{-t} \left (7 e^{2 t}+6 e^t \cos (3 t)+7\right )-\frac {7}{200} e^{-t} \left (e^{2 t}-2 e^t \cos (3 t)+1\right ) \left (3 e^{-t} \left (e^{2 t} (t-1)+t+1\right )-\frac {2}{9} \sin (3 t)+\frac {2}{3} t \cos (3 t)\right )+\frac {1}{200} e^{-t} \left (7 e^{2 t}+6 e^t \cos (3 t)+7\right ) \left (-\frac {7}{3} e^t \left (3 e^{-2 t} t+3 t+3 e^{-2 t}-3\right )-\frac {2}{9} \sin (3 t)+\frac {2}{3} t \cos (3 t)\right )+\frac {7}{200} e^{-t} \left (3 e^{2 t}-2 e^t \sin (3 t)-3\right ) \left (e^{-t} \left (e^{2 t} (t-1)-t-1\right )-\frac {2}{3} t \sin (3 t)-\frac {2}{9} \cos (3 t)\right )+\frac {1}{200} e^{-t} \left (7 e^{2 t}+2 e^t \sin (3 t)-7\right ) \left (e^{-t} \left (7 e^{2 t} (t-1)-7 (t+1)\right )+2 t \sin (3 t)+\frac {2}{3} \cos (3 t)\right )\right \}\right \} \]
Maple: cpu = 0.047 (sec), leaf count = 64 \[ \left \{ \left \{ x \left ( t \right ) ={\frac {14\,t}{9}}+{\it \_C1}\,{ {\rm e}^{t}}+{\it \_C2}\,\cos \left ( 3\,t \right ) +{\it \_C3}\,{ {\rm e}^{-t}}+{\it \_C4}\,\sin \left ( 3\,t \right ) ,y \left ( t \right ) =-{\it \_C1}\,{{\rm e}^{t}}+{\frac {3\,{\it \_C2}\,\cos \left ( 3\,t \right ) }{7}}-{\it \_C3}\,{{\rm e}^{-t}}+{\frac {3\,{\it \_C4}\,\sin \left ( 3\,t \right ) }{7}}-{\frac {4\,t}{3}} \right \} \right \} \]