\[ \boxed { \left \{ a{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +b{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =\alpha \,x \left ( t \right ) +\beta \,y \left ( t \right ) ,b{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -a{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =\beta \,x \left ( t \right ) -\alpha \,y \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.011001 (sec), leaf count = 183 \[ \left \{\left \{x(t)\to c_2 e^{\frac {t (a \alpha +b \beta )}{a^2+b^2}} \sin \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )+c_1 e^{\frac {t (a \alpha +b \beta )}{a^2+b^2}} \cos \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right ),y(t)\to c_2 e^{\frac {t (a \alpha +b \beta )}{a^2+b^2}} \cos \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )-c_1 e^{\frac {t (a \alpha +b \beta )}{a^2+b^2}} \sin \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )\right \}\right \} \]
Maple: cpu = 0.062 (sec), leaf count = 144 \[ \left \{ \left \{ x \left ( t \right ) ={\it \_C1}\,{{\rm e}^{{\frac { \left ( ia\beta -i\alpha \,b+a\alpha +\beta \,b \right ) t}{{a}^{2}+{b}^{2} }}}}+{\it \_C2}\,{{\rm e}^{-{\frac { \left ( ia\beta -i\alpha \,b-a\alpha -\beta \,b \right ) t}{{a}^{2}+{b}^{2}}}}},y \left ( t \right ) =i \left ( {\it \_C1}\,{{\rm e}^{{\frac { \left ( ia\beta -i\alpha \,b+a\alpha +\beta \,b \right ) t}{{a}^{2}+{b}^{2}}}}}-{\it \_C2}\,{{\rm e}^{-{\frac { \left ( ia\beta -i\alpha \,b-a\alpha -\beta \,b \right ) t}{{a}^{2}+{b}^{2} }}}} \right ) \right \} \right \} \]