\[ \boxed { \left \{ 2\,{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) +{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) =2\,t,{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) +{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =\sinh \left ( 2\,t \right ) \right \} } \]
Mathematica: cpu = 0.114015 (sec), leaf count = 284 \[ \left \{\left \{x(t)\to \frac {1}{4} c_4 e^{-2 t} \left (2 e^{2 t} t-e^{2 t}+1\right )+c_2 t+c_1+t \left (\frac {t^2}{2}+\frac {t}{2}-\frac {e^{4 t}}{8}+e^{2 t} \left (\frac {t}{2}-\frac {1}{4}\right )\right )+\frac {1}{48} e^{-2 t} \left (-4 e^{2 t} \left (4 t^2-3 t+3\right ) t-12 e^{4 t} t+3 e^{6 t}-6\right )+\frac {1}{4} e^{-2 t} \left (-2 e^{2 t} \left (\frac {t}{2}-\frac {1}{4}\right )+\frac {e^{4 t}}{4}-t\right ) \left (2 e^{2 t} t-e^{2 t}+1\right ),y(t)\to \frac {1}{2} c_4 e^{-2 t} \left (e^{2 t}-1\right )+c_3+\frac {1}{2} e^{-2 t} \left (e^{2 t}-1\right ) \left (-2 e^{2 t} \left (\frac {t}{2}-\frac {1}{4}\right )+\frac {e^{4 t}}{4}-t\right )+\frac {1}{8} e^{-2 t} \left (4 e^{4 t} t-4 e^{2 t} (t-1) t-e^{6 t}+2\right )\right \}\right \} \]
Maple: cpu = 0.110 (sec), leaf count = 98 \[ \left \{ \left \{ x \left ( t \right ) =-{\frac {\sinh \left ( 2\,t \right ) }{16}}-{\frac {\cosh \left ( 2\,t \right ) }{16}}-{\frac {5\,{ {\rm e}^{-2\,t}}}{16}}-{\frac {{{\rm e}^{-2\,t}}t}{4}}+{\frac {{t}^{3} }{6}}+{\frac {{{\rm e}^{-2\,t}}{\it \_C2}}{4}}+{\frac {{t}^{2}}{4}}+{ \it \_C3}\,t+{\it \_C4},y \left ( t \right ) ={\frac {3\,\cosh \left ( 2 \,t \right ) }{8}}-{\frac {\sinh \left ( 2\,t \right ) }{8}}+{\frac {3\,{ {\rm e}^{-2\,t}}}{8}}+{\frac {{{\rm e}^{-2\,t}}t}{2}}-{\frac {{t}^{2} }{2}}-{\frac {{{\rm e}^{-2\,t}}{\it \_C2}}{2}}+{\frac {t}{2}}+{\it \_C3}+{\it \_C1} \right \} \right \} \]