\[ \boxed { \left \{ 3\,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +7\,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +8\,x \left ( t \right ) +24\,y \left ( t \right ) ={{\rm e}^{2\,t}},4\,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +9\,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +11\,x \left ( t \right ) +31\,y \left ( t \right ) ={{\rm e}^{t}} \right \} } \]
Mathematica: cpu = 0.059508 (sec), leaf count = 162 \[ \left \{\left \{x(t)\to -c_1 e^{-4 t} (t-1)-c_2 e^{-4 t} t-e^t t \left (-\frac {4 t}{5}+\frac {1}{36} e^t (30 t+19)-\frac {11}{25}\right )-e^t (t-1) \left (\frac {4 t}{5}-\frac {1}{36} e^t (30 t+49)+\frac {31}{25}\right ),y(t)\to c_1 e^{-4 t} t+c_2 e^{-4 t} (t+1)+e^t (t+1) \left (-\frac {4 t}{5}+\frac {1}{36} e^t (30 t+19)-\frac {11}{25}\right )+e^t t \left (\frac {4 t}{5}-\frac {1}{36} e^t (30 t+49)+\frac {31}{25}\right )\right \}\right \} \]
Maple: cpu = 0.047 (sec), leaf count = 65 \[ \left \{ \left \{ x \left ( t \right ) ={{\rm e}^{-4\,t}}{\it \_C2}+{ {\rm e}^{-4\,t}}t{\it \_C1}-{\frac {49\,{{\rm e}^{2\,t}}}{36}}+{\frac {31\,{{\rm e}^{t}}}{25}},y \left ( t \right ) ={\frac {19\,{{\rm e}^{2\, t}}}{36}}-{{\rm e}^{-4\,t}}{\it \_C2}-{{\rm e}^{-4\,t}}t{\it \_C1}-{ {\rm e}^{-4\,t}}{\it \_C1}-{\frac {11\,{{\rm e}^{t}}}{25}} \right \} \right \} \]