\[ \boxed { \left \{ {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) +{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) -x \left ( t \right ) =0,{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) -{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =0 \right \} } \]
Mathematica: cpu = 0.037505 (sec), leaf count = 420 \[ \left \{\left \{x(t)\to -\frac {1}{5} c_1 e^{\frac {t}{2}-\frac {\sqrt {5} t}{2}} \left (\sqrt {5} e^{\sqrt {5} t}-5 e^{\frac {\sqrt {5} t}{2}+\frac {t}{2}}-\sqrt {5}\right )+\frac {c_2 e^{\frac {t}{2}-\frac {\sqrt {5} t}{2}} \left (e^{\sqrt {5} t}-1\right )}{\sqrt {5}}-\frac {1}{10} c_4 e^{\frac {t}{2}-\frac {\sqrt {5} t}{2}} \left (5 e^{\sqrt {5} t}+\sqrt {5} e^{\sqrt {5} t}-10 e^{\frac {\sqrt {5} t}{2}+\frac {t}{2}}+5-\sqrt {5}\right ),y(t)\to -\frac {1}{10} c_1 e^{-\frac {\sqrt {5} t}{2}} \left (-5 e^{t/2}-\sqrt {5} e^{t/2}+10 e^{\frac {\sqrt {5} t}{2}}-5 e^{\sqrt {5} t+\frac {t}{2}}+\sqrt {5} e^{\sqrt {5} t+\frac {t}{2}}\right )+\frac {1}{10} c_2 e^{-\frac {\sqrt {5} t}{2}} \left (-5 e^{t/2}-\sqrt {5} e^{t/2}+10 e^{\frac {\sqrt {5} t}{2}}-5 e^{\sqrt {5} t+\frac {t}{2}}+\sqrt {5} e^{\sqrt {5} t+\frac {t}{2}}\right )+\frac {c_4 e^{\frac {t}{2}-\frac {\sqrt {5} t}{2}} \left (e^{\sqrt {5} t}-1\right )}{\sqrt {5}}+c_3\right \}\right \} \]
Maple: cpu = 0.063 (sec), leaf count = 73 \[ \left \{ \left \{ x \left ( t \right ) = \left ( -{\frac {1}{2}}-{\frac { \sqrt {5}}{2}} \right ) {\it \_C3}\,{{\rm e}^{{\frac { \left ( \sqrt {5} +1 \right ) t}{2}}}}+ \left ( {\frac {\sqrt {5}}{2}}-{\frac {1}{2}} \right ) {\it \_C4}\,{{\rm e}^{-{\frac { \left ( \sqrt {5}-1 \right ) t }{2}}}}+{\it \_C1}\,{{\rm e}^{t}},y \left ( t \right ) ={\it \_C2}+{\it \_C3}\,{{\rm e}^{{\frac { \left ( \sqrt {5}+1 \right ) t}{2}}}}+{\it \_C4}\,{{\rm e}^{-{\frac { \left ( \sqrt {5}-1 \right ) t}{2}}}} \right \} \right \} \]