\[ \boxed { \left \{ 3\,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +7\,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +34\,x \left ( t \right ) +38\,y \left ( t \right ) ={{\rm e}^{t}},4\,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +9\,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +44\,x \left ( t \right ) +49\,y \left ( t \right ) =t \right \} } \]
Mathematica: cpu = 0.045006 (sec), leaf count = 322 \[ \left \{\left \{x(t)\to \frac {1}{5} c_1 e^{-6 t} \left (4 e^{5 t}+1\right )-\frac {1}{5} c_2 e^{-6 t} \left (e^{5 t}-1\right )-\frac {1}{5} e^{-6 t} \left (e^{5 t}-1\right ) \left (\frac {16}{5} e^{6 t} \left (\frac {t}{6}-\frac {1}{36}\right )+4 e^{2 t}-\frac {4 e^{7 t}}{7}-\frac {31}{5} e^t (t-1)\right )+\frac {1}{25} e^{-6 t} \left (4 e^{5 t}+1\right ) \left (e^{6 t} \left (\frac {2 t}{3}-\frac {1}{9}\right )-20 e^{2 t}-\frac {5 e^{7 t}}{7}+e^t (31 t-31)\right ),y(t)\to -\frac {4}{5} c_1 e^{-6 t} \left (e^{5 t}-1\right )+\frac {1}{5} c_2 e^{-6 t} \left (e^{5 t}+4\right )+\frac {1}{5} e^{-6 t} \left (e^{5 t}+4\right ) \left (\frac {16}{5} e^{6 t} \left (\frac {t}{6}-\frac {1}{36}\right )+4 e^{2 t}-\frac {4 e^{7 t}}{7}-\frac {31}{5} e^t (t-1)\right )-\frac {4}{25} e^{-6 t} \left (e^{5 t}-1\right ) \left (e^{6 t} \left (\frac {2 t}{3}-\frac {1}{9}\right )-20 e^{2 t}-\frac {5 e^{7 t}}{7}+e^t (31 t-31)\right )\right \}\right \} \]
Maple: cpu = 0.047 (sec), leaf count = 52 \[ \left \{ \left \{ x \left ( t \right ) ={{\rm e}^{-6\,t}}{\it \_C2}+{ \it \_C1}\,{{\rm e}^{-t}}-{\frac {56}{9}}+{\frac {19\,t}{3}}-{\frac { 29\,{{\rm e}^{t}}}{7}},y \left ( t \right ) =4\,{{\rm e}^{-6\,t}}{\it \_C2}-{\it \_C1}\,{{\rm e}^{-t}}+{\frac {55}{9}}+{\frac {24\,{{\rm e}^ {t}}}{7}}-{\frac {17\,t}{3}} \right \} \right \} \]