\[ \boxed { \left \{ 3\,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +7\,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +x \left ( t \right ) +24\,y \left ( t \right ) =3,4\,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +9\,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +2\,x \left ( t \right ) +31\,y \left ( t \right ) ={{\rm e}^{t}} \right \} } \]
Mathematica: cpu = 0.161521 (sec), leaf count = 180 \[ \left \{\left \{x(t)\to -c_2 e^{-4 t} \sin (t)+c_1 e^{-4 t} (\cos (t)-\sin (t))+\frac {1}{442} \left (3 \left (153 e^t-754\right ) \sin (t)+31 \left (17 e^t-78\right ) \cos (t)\right ) (\cos (t)-\sin (t))+\frac {1}{221} \sin (t) \left (\left (493 e^t-2340\right ) \sin (t)+\left (34 e^t-78\right ) \cos (t)\right ),y(t)\to 2 c_1 e^{-4 t} \sin (t)+c_2 e^{-4 t} (\sin (t)+\cos (t))+\frac {1}{221} \sin (t) \left (3 \left (153 e^t-754\right ) \sin (t)+31 \left (17 e^t-78\right ) \cos (t)\right )-\frac {1}{221} (\sin (t)+\cos (t)) \left (\left (493 e^t-2340\right ) \sin (t)+\left (34 e^t-78\right ) \cos (t)\right )\right \}\right \} \]
Maple: cpu = 0.047 (sec), leaf count = 71 \[ \left \{ \left \{ x \left ( t \right ) ={{\rm e}^{-4\,t}}\sin \left ( t \right ) {\it \_C2}+{{\rm e}^{-4\,t}}\cos \left ( t \right ) {\it \_C1}- {\frac {93}{17}}+{\frac {31\,{{\rm e}^{t}}}{26}},y \left ( t \right ) =- {{\rm e}^{-4\,t}}\sin \left ( t \right ) {\it \_C2}-{{\rm e}^{-4\,t}} \cos \left ( t \right ) {\it \_C2}-{{\rm e}^{-4\,t}}\cos \left ( t \right ) {\it \_C1}+{{\rm e}^{-4\,t}}\sin \left ( t \right ) {\it \_C1}- {\frac {2\,{{\rm e}^{t}}}{13}}+{\frac {6}{17}} \right \} \right \} \]