\[ \boxed { \left \{ {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) -2\,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =2\,t-\cos \left ( 2\,t \right ) ,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -x \left ( t \right ) +2\,y \left ( t \right ) =0 \right \} } \]
Mathematica: cpu = 0.180523 (sec), leaf count = 226 \[ \left \{\left \{x(t)\to 7 \left (c_2+t^2-\frac {1}{2} \sin (2 t)\right )+8 \left (c_1 e^{t/2}+c_2 \left (e^{t/2}-1\right )+\frac {1}{68} e^{-t/2} \left (e^{t/2} \cos (2 t)-2 \left (17 \left (2 e^{t/2} t^2+e^{t/2} (t+2)\right )-15 e^{t/2} \sin (2 t)\right )\right )\right ),y(t)\to \frac {3}{2} \left (c_2+t^2-\frac {1}{2} \sin (2 t)\right )+2 \left (c_1 e^{t/2}+c_2 \left (e^{t/2}-1\right )+\frac {1}{68} e^{-t/2} \left (e^{t/2} \cos (2 t)-2 \left (17 \left (2 e^{t/2} t^2+e^{t/2} (t+2)\right )-15 e^{t/2} \sin (2 t)\right )\right )\right )\right \}\right \} \]
Maple: cpu = 0.078 (sec), leaf count = 69 \[ \left \{ \left \{ x \left ( t \right ) ={\frac {\sin \left ( 2\,t \right ) }{34}}+{\frac {2\,\cos \left ( 2\,t \right ) }{17}}-{t}^{2}+2\, {\it \_C1}\,{{\rm e}^{t/2}}-4\,t+{\it \_C2},y \left ( t \right ) ={ \frac {\cos \left ( 2\,t \right ) }{34}}+{\frac {9\,\sin \left ( 2\,t \right ) }{68}}-t+{\frac {{\it \_C1}}{2}{{\rm e}^{{\frac {t}{2}}}}}+2- {\frac {{t}^{2}}{2}}+{\frac {{\it \_C2}}{2}} \right \} \right \} \]