\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +x \left ( t \right ) =2\,t,{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) -9\,x \left ( t \right ) +3\,y \left ( t \right ) =\sin \left ( 2\,t \right ) \right \} } \]
Mathematica: cpu = 0.590075 (sec), leaf count = 614 \[ \left \{\left \{x(t)\to \frac {1}{16} c_1 e^{-3 t} \left (20 e^{4 t} t+7 e^{4 t}+9\right )+\frac {1}{16} c_2 e^{-3 t} \left (4 e^{4 t} t+3 e^{4 t}-3\right )-\frac {3}{16} c_3 e^{-3 t} \left (4 e^{4 t} t-e^{4 t}+1\right )+\frac {e^{-4 t} \left (20 e^{4 t} t+7 e^{4 t}+9\right ) \left (10400 \left (t^2+2 t+2\right )+\left (260 t-225 e^{4 t}-351\right ) \sin (2 t)+2 \left (260 t+75 e^{4 t}-91\right ) \cos (2 t)\right )}{83200}-\frac {3 e^{-4 t} \left (4 e^{4 t} t-e^{4 t}+1\right ) \left (5200 \left (2 t^2+5 t+5\right )+\left (260 t-75 e^{4 t}-221\right ) \sin (2 t)+\left (520 t+50 e^{4 t}+78\right ) \cos (2 t)\right )}{41600}+\frac {e^{-4 t} \left (4 e^{4 t} t+3 e^{4 t}-3\right ) \left (130 \left (40 \left (t^2+t+1\right )+t \sin (2 t)\right )+\left (260 t-225 e^{4 t}-351\right ) \cos (2 t)+\left (675 e^{4 t}-611\right ) \sin (t) \cos (t)\right )}{41600},y(t)\to \frac {1}{8} c_1 e^{-3 t} \left (20 e^{4 t} t-3 e^{4 t}+3\right )+\frac {1}{8} c_2 e^{-3 t} \left (4 e^{4 t} t+e^{4 t}-1\right )-\frac {1}{8} c_3 e^{-3 t} \left (12 e^{4 t} t-9 e^{4 t}+1\right )+\frac {e^{-4 t} \left (20 e^{4 t} t-3 e^{4 t}+3\right ) \left (10400 \left (t^2+2 t+2\right )+\left (260 t-225 e^{4 t}-351\right ) \sin (2 t)+2 \left (260 t+75 e^{4 t}-91\right ) \cos (2 t)\right )}{41600}-\frac {e^{-4 t} \left (12 e^{4 t} t-9 e^{4 t}+1\right ) \left (5200 \left (2 t^2+5 t+5\right )+\left (260 t-75 e^{4 t}-221\right ) \sin (2 t)+\left (520 t+50 e^{4 t}+78\right ) \cos (2 t)\right )}{20800}+\frac {e^{-4 t} \left (4 e^{4 t} t+e^{4 t}-1\right ) \left (130 \left (40 \left (t^2+t+1\right )+t \sin (2 t)\right )+\left (260 t-225 e^{4 t}-351\right ) \cos (2 t)+\left (675 e^{4 t}-611\right ) \sin (t) \cos (t)\right )}{20800}\right \}\right \} \]
Maple: cpu = 0.094 (sec), leaf count = 80 \[ \left \{ \left \{ x \left ( t \right ) =-{\frac {2\,\cos \left ( 2\,t \right ) }{325}}+4-{\frac {36\,\sin \left ( 2\,t \right ) }{325}}+2\,t+{ \it \_C1}\,{{\rm e}^{t}}+{\it \_C2}\,{{\rm e}^{-3\,t}}+{\it \_C3}\,{ {\rm e}^{t}}t,y \left ( t \right ) ={\frac {16\,\cos \left ( 2\,t \right ) }{325}}-{\frac {37\,\sin \left ( 2\,t \right ) }{325}}+2\,{\it \_C1}\,{{\rm e}^{t}}+{\frac {2\,{\it \_C2}\,{{\rm e}^{-3\,t}}}{3}}+2\, {\it \_C3}\,{{\rm e}^{t}}t-{\it \_C3}\,{{\rm e}^{t}}+10+6\,t \right \} \right \} \]