\[ \boxed { \left \{ {\frac {{\rm d}^{3}}{{\rm d}{t}^{3}}}y \left ( t \right ) -{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) +2\,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -x \left ( t \right ) =t,{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) -2\,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +y \left ( t \right ) =0 \right \} } \]
Mathematica: cpu = 0.197525 (sec), leaf count = 1132 \[ \left \{\left \{x(t)\to \frac {1}{64} e^{-t} \left (2 e^{2 t} t^2-6 e^{2 t} t+7 e^{2 t}+1\right ) \left (e^t (1-t)+e^{-t} \left (-2 t^3-8 t^2-17 t-17\right )\right )+\frac {1}{64} e^{-t} \left (2 e^{2 t} t^2+6 e^{2 t} t+e^{2 t}-1\right ) \left (e^t (t-1)+e^{-t} \left (-2 t^3-4 t^2-7 t-7\right )\right )-\frac {1}{384} e^{-t} \left (2 e^{2 t} t^2+2 e^{2 t} t-e^{2 t}+1\right ) \left (e^t (9 t-9)+e^{-t} \left (-4 t^4-22 t^3-48 t^2-87 t-87\right )\right )+\frac {1}{384} e^{-t} \left (2 e^{2 t} t^2-2 e^{2 t} t+e^{2 t}-1\right ) \left (e^t (9 t-9)+e^{-t} \left (-4 t^4+2 t^3+24 t^2+9 t+9\right )\right )+\frac {1}{192} e^{-t} \left (2 e^{2 t} t-e^{2 t}+1\right ) \left (-e^t (9 t-9)-e^{-t} \left (4 t^4+10 t^3+9 t+9\right )\right )+\frac {1}{8} e^{-t} \left (2 e^{2 t} t^2-6 e^{2 t} t+7 e^{2 t}+1\right ) c_1+\frac {1}{8} e^{-t} \left (2 e^{2 t} t^2+6 e^{2 t} t+e^{2 t}-1\right ) c_2-\frac {1}{8} e^{-t} \left (2 e^{2 t} t^2+2 e^{2 t} t-e^{2 t}+1\right ) c_3+\frac {1}{4} e^{-t} \left (2 e^{2 t} t-e^{2 t}+1\right ) c_4+\frac {1}{8} e^{-t} \left (2 e^{2 t} t^2-2 e^{2 t} t+e^{2 t}-1\right ) c_5,y(t)\to -\frac {1}{384} e^{-t} \left (4 e^{2 t} t^3-18 e^{2 t} t^2+18 e^{2 t} t-9 e^{2 t}+9\right ) \left (e^t (1-t)+e^{-t} \left (-2 t^3-8 t^2-17 t-17\right )\right )-\frac {1}{384} e^{-t} \left (4 e^{2 t} t^3+18 e^{2 t} t^2-18 e^{2 t} t+9 e^{2 t}-9\right ) \left (e^t (t-1)+e^{-t} \left (-2 t^3-4 t^2-7 t-7\right )\right )+\frac {e^{-t} \left (4 e^{2 t} t^3+6 e^{2 t} t^2-30 e^{2 t} t+39 e^{2 t}+9\right ) \left (e^t (9 t-9)+e^{-t} \left (-4 t^4-22 t^3-48 t^2-87 t-87\right )\right )}{2304}-\frac {e^{-t} \left (4 e^{2 t} t^3-6 e^{2 t} t^2-18 e^{2 t} t+9 e^{2 t}-9\right ) \left (e^t (9 t-9)+e^{-t} \left (-4 t^4+2 t^3+24 t^2+9 t+9\right )\right )}{2304}-\frac {1}{384} e^{-t} \left (2 e^{2 t} t^2-2 e^{2 t} t-3 e^{2 t}+3\right ) \left (-e^t (9 t-9)-e^{-t} \left (4 t^4+10 t^3+9 t+9\right )\right )-\frac {1}{48} e^{-t} \left (4 e^{2 t} t^3-18 e^{2 t} t^2+18 e^{2 t} t-9 e^{2 t}+9\right ) c_1-\frac {1}{48} e^{-t} \left (4 e^{2 t} t^3+18 e^{2 t} t^2-18 e^{2 t} t+9 e^{2 t}-9\right ) c_2+\frac {1}{48} e^{-t} \left (4 e^{2 t} t^3+6 e^{2 t} t^2-30 e^{2 t} t+39 e^{2 t}+9\right ) c_3-\frac {1}{8} e^{-t} \left (2 e^{2 t} t^2-2 e^{2 t} t-3 e^{2 t}+3\right ) c_4-\frac {1}{48} e^{-t} \left (4 e^{2 t} t^3-6 e^{2 t} t^2-18 e^{2 t} t+9 e^{2 t}-9\right ) c_5\right \}\right \} \]
Maple: cpu = 0.031 (sec), leaf count = 75 \[ \left \{ \left \{ x \left ( t \right ) =-2-t-{\it \_C3}\,{{\rm e}^{t}}-6 \,{\it \_C5}\,{{\rm e}^{t}}-{\frac {2\,{\it \_C2}\,{{\rm e}^{-t}}}{3}} -2\,{\it \_C4}\,{{\rm e}^{t}}t-3\,{\it \_C5}\,{{\rm e}^{t}}{t}^{2},y \left ( t \right ) =-2+{\it \_C1}\,{{\rm e}^{t}}+{\it \_C2}\,{{\rm e}^{ -t}}+{\it \_C3}\,{{\rm e}^{t}}t+{\it \_C4}\,{{\rm e}^{t}}{t}^{2}+{\it \_C5}\,{{\rm e}^{t}}{t}^{3} \right \} \right \} \]