\[ \boxed { \left \{ {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) +x \left ( t \right ) +y \left ( t \right ) =-5,{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) -4\,x \left ( t \right ) -3\,y \left ( t \right ) =-3 \right \} } \]
Mathematica: cpu = 0.103513 (sec), leaf count = 554 \[ \left \{\left \{x(t)\to -\frac {1}{4} c_4 e^{-t} \left (e^{2 t} t+t-e^{2 t}+1\right )-\frac {1}{2} c_1 e^{-t} \left (e^{2 t} t-t-e^{2 t}-1\right )-\frac {1}{2} c_2 e^{-t} \left (e^{2 t} t+t-2 e^{2 t}+2\right )-\frac {1}{4} c_3 e^{-t} \left (e^{2 t}-1\right ) t-\frac {1}{8} e^{-t} \left (e^{-t} (-13 t-10)+e^t (10-13 t)\right ) \left (e^{2 t} t+t-e^{2 t}+1\right )-\frac {1}{8} e^{-t} \left (e^{2 t}-1\right ) t \left (e^{-t} (-13 t-23)+e^t (13 t-23)\right )-\frac {1}{8} e^{-t} \left (e^{2 t} t+t-2 e^{2 t}+2\right ) \left (e^t (13 t-23)+e^{-t} (13 t+23)\right )-\frac {1}{8} e^{-t} \left (e^{2 t} t-t-e^{2 t}-1\right ) \left (e^t (36-13 t)+e^{-t} (13 t+36)\right ),y(t)\to c_1 e^{-t} \left (e^{2 t}-1\right ) t+\frac {1}{2} c_4 e^{-t} \left (e^{2 t}+1\right ) t+c_2 e^{-t} \left (e^{2 t} t+t-e^{2 t}+1\right )+\frac {1}{2} c_3 e^{-t} \left (e^{2 t} t-t+e^{2 t}+1\right )+\frac {1}{4} e^{-t} \left (e^{2 t}+1\right ) \left (e^{-t} (-13 t-10)+e^t (10-13 t)\right ) t+\frac {1}{4} e^{-t} \left (e^{2 t}-1\right ) \left (e^t (36-13 t)+e^{-t} (13 t+36)\right ) t+\frac {1}{4} e^{-t} \left (e^{2 t} t-t+e^{2 t}+1\right ) \left (e^{-t} (-13 t-23)+e^t (13 t-23)\right )+\frac {1}{4} e^{-t} \left (e^{2 t} t+t-e^{2 t}+1\right ) \left (e^t (13 t-23)+e^{-t} (13 t+23)\right )\right \}\right \} \]
Maple: cpu = 0.032 (sec), leaf count = 72 \[ \left \{ \left \{ x \left ( t \right ) =18+{\it \_C1}\,{{\rm e}^{t}}+{ \it \_C2}\,{{\rm e}^{-t}}+{\it \_C3}\,{{\rm e}^{t}}t+{\it \_C4}\,{ {\rm e}^{-t}}t,y \left ( t \right ) =-2\,{\it \_C1}\,{{\rm e}^{t}}-2\,{ \it \_C2}\,{{\rm e}^{-t}}-2\,{\it \_C3}\,{{\rm e}^{t}}t-2\,{\it \_C3} \,{{\rm e}^{t}}-2\,{\it \_C4}\,{{\rm e}^{-t}}t+2\,{\it \_C4}\,{{\rm e} ^{-t}}-23 \right \} \right \} \]