\[ \boxed { \left \{ {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) -2\,y \left ( t \right ) ={{\rm e}^{2\,t}},2\,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) -3\,x \left ( t \right ) =0 \right \} } \]
Mathematica: cpu = 0.419553 (sec), leaf count = 928 \[ \left \{\left \{x(t)\to \frac {1}{46} e^{t/2} c_1 \left (23 \cos \left (\frac {\sqrt {23} t}{2}\right )+23 e^{t/2}-3 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )\right )+\frac {e^{3 t/2} \left (23 e^{t/2} \cos \left (\frac {\sqrt {23} t}{2}\right )-7 \sqrt {23} e^{t/2} \sin \left (\frac {\sqrt {23} t}{2}\right )+46\right ) \left (23 \cos \left (\frac {\sqrt {23} t}{2}\right )+23 e^{t/2}-3 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )\right )}{12696}+\frac {1}{69} e^{t/2} c_3 \left (-23 \cos \left (\frac {\sqrt {23} t}{2}\right )+23 e^{t/2}-\sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )\right )+\frac {1}{138} e^{t/2} c_2 \left (-23 \cos \left (\frac {\sqrt {23} t}{2}\right )+23 e^{t/2}+11 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )\right )+\frac {e^{3 t/2} \left (-23 \cos \left (\frac {\sqrt {23} t}{2}\right )+23 e^{t/2}+11 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )\right ) \left (46 e^{t/2} \cos \left (\frac {\sqrt {23} t}{2}\right )+4 \sqrt {23} e^{t/2} \sin \left (\frac {\sqrt {23} t}{2}\right )+23\right )}{19044}+\frac {e^{3 t/2} \left (-23 \cos \left (\frac {\sqrt {23} t}{2}\right )+23 e^{t/2}-\sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )\right ) \left (-161 e^{t/2} \cos \left (\frac {\sqrt {23} t}{2}\right )+13 \sqrt {23} e^{t/2} \sin \left (\frac {\sqrt {23} t}{2}\right )+92\right )}{38088},y(t)\to \frac {1}{138} e^{t/2} c_2 \left (-23 \cos \left (\frac {\sqrt {23} t}{2}\right )+23 e^{t/2}-25 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )\right )+\frac {e^{3 t/2} \left (46 e^{t/2} \cos \left (\frac {\sqrt {23} t}{2}\right )+4 \sqrt {23} e^{t/2} \sin \left (\frac {\sqrt {23} t}{2}\right )+23\right ) \left (-23 \cos \left (\frac {\sqrt {23} t}{2}\right )+23 e^{t/2}-25 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )\right )}{19044}+\frac {1}{69} e^{t/2} c_3 \left (46 \cos \left (\frac {\sqrt {23} t}{2}\right )+23 e^{t/2}-4 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )\right )+\frac {1}{46} e^{t/2} c_1 \left (-23 \cos \left (\frac {\sqrt {23} t}{2}\right )+23 e^{t/2}+11 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )\right )+\frac {e^{3 t/2} \left (-23 \cos \left (\frac {\sqrt {23} t}{2}\right )+23 e^{t/2}+11 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )\right ) \left (23 e^{t/2} \cos \left (\frac {\sqrt {23} t}{2}\right )-7 \sqrt {23} e^{t/2} \sin \left (\frac {\sqrt {23} t}{2}\right )+46\right )}{12696}+\frac {e^{3 t/2} \left (46 \cos \left (\frac {\sqrt {23} t}{2}\right )+23 e^{t/2}-4 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )\right ) \left (-161 e^{t/2} \cos \left (\frac {\sqrt {23} t}{2}\right )+13 \sqrt {23} e^{t/2} \sin \left (\frac {\sqrt {23} t}{2}\right )+92\right )}{38088}\right \}\right \} \]
Maple: cpu = 0.047 (sec), leaf count = 118 \[ \left \{ \left \{ x \left ( t \right ) ={\frac {{{\rm e}^{2\,t}}}{4}}+{ \it \_C1}\,{{\rm e}^{t}}+{\it \_C2}\,{{\rm e}^{{\frac {t}{2}}}}\cos \left ( {\frac {\sqrt {23}t}{2}} \right ) +{\it \_C3}\,{{\rm e}^{{ \frac {t}{2}}}}\sin \left ( {\frac {\sqrt {23}t}{2}} \right ) ,y \left ( t \right ) =-{\frac {{{\rm e}^{2\,t}}}{8}}+{\it \_C1}\,{{\rm e}^{t}}-{ \frac {7\,{\it \_C2}}{4}{{\rm e}^{{\frac {t}{2}}}}\cos \left ( {\frac { \sqrt {23}t}{2}} \right ) }+{\frac {{\it \_C2}\,\sqrt {23}}{4}{{\rm e}^ {{\frac {t}{2}}}}\sin \left ( {\frac {\sqrt {23}t}{2}} \right ) }-{ \frac {7\,{\it \_C3}}{4}{{\rm e}^{{\frac {t}{2}}}}\sin \left ( {\frac { \sqrt {23}t}{2}} \right ) }-{\frac {{\it \_C3}\,\sqrt {23}}{4}{{\rm e}^ {{\frac {t}{2}}}}\cos \left ( {\frac {\sqrt {23}t}{2}} \right ) } \right \} \right \} \]