\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +y \left ( t \right ) =f \left ( t \right ) ,{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) +{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +x \left ( t \right ) +y \left ( t \right ) =g \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.058507 (sec), leaf count = 44 \[ \left \{\left \{x(t)\to -f''(t)-f'(t)-f(t)+g'(t)+g(t),y(t)\to f''(t)+f(t)-g'(t)\right \}\right \} \]
Maple: cpu = 0.015 (sec), leaf count = 48 \[ \left \{ \left \{ x \left ( t \right ) =-{\frac {\rm d}{{\rm d}t}}f \left ( t \right ) +g \left ( t \right ) -f \left ( t \right ) -{\frac { {\rm d}^{2}}{{\rm d}{t}^{2}}}f \left ( t \right ) +{\frac {\rm d}{ {\rm d}t}}g \left ( t \right ) ,y \left ( t \right ) =f \left ( t \right ) + {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}f \left ( t \right ) -{\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right \} \right \} \]