\[ \left \{a x'(t)+b y'(t)=\alpha x(t)+\beta y(t),b x'(t)-a y'(t)=\beta x(t)-\alpha y(t)\right \} \] ✓ Mathematica : cpu = 0.0138732 (sec), leaf count = 145
\[\left \{\left \{x(t)\to e^{\frac {t (a \alpha +b \beta )}{a^2+b^2}} \left (c_2 \sin \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )+c_1 \cos \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )\right ),y(t)\to e^{\frac {t (a \alpha +b \beta )}{a^2+b^2}} \left (c_2 \cos \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )-c_1 \sin \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )\right )\right \}\right \}\]
✓ Maple : cpu = 0.116 (sec), leaf count = 152
\[ \left \{ \left \{ x \left ( t \right ) ={\it \_C1}\,{{\rm e}^{{\frac {t \left ( \left ( i\beta +\alpha \right ) a- \left ( i\alpha -\beta \right ) b \right ) }{{a}^{2}+{b}^{2}}}}}+{\it \_C2}\,{{\rm e}^{-{\frac { \left ( \left ( i\beta -\alpha \right ) a- \left ( i\alpha +\beta \right ) b \right ) t}{{a}^{2}+{b}^{2}}}}},y \left ( t \right ) =i \left ( {\it \_C1}\,{{\rm e}^{{\frac {t \left ( \left ( i\beta +\alpha \right ) a- \left ( i\alpha -\beta \right ) b \right ) }{{a}^{2}+{b}^{2}}}}}-{\it \_C2}\,{{\rm e}^{-{\frac { \left ( \left ( i\beta -\alpha \right ) a- \left ( i\alpha +\beta \right ) b \right ) t}{{a}^{2}+{b}^{2}}}}} \right ) \right \} \right \} \]