\[ \left \{a x'(t)+b y'(t)=\alpha x(t)+\beta y(t),b x'(t)-a y'(t)=\beta x(t)-\alpha y(t)\right \} \] ✓ Mathematica : cpu = 0.0089771 (sec), leaf count = 183
DSolve[{a*Derivative[1][x][t] + b*Derivative[1][y][t] == alpha*x[t] + beta*y[t], b*Derivative[1][x][t] - a*Derivative[1][y][t] == beta*x[t] - alpha*y[t]},{x[t], y[t]},t]
\[\left \{\left \{x(t)\to c_1 e^{\frac {t (a \alpha +b \beta )}{a^2+b^2}} \cos \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )+c_2 e^{\frac {t (a \alpha +b \beta )}{a^2+b^2}} \sin \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right ),y(t)\to c_2 e^{\frac {t (a \alpha +b \beta )}{a^2+b^2}} \cos \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )-c_1 e^{\frac {t (a \alpha +b \beta )}{a^2+b^2}} \sin \left (\frac {t (a \beta -\alpha b)}{a^2+b^2}\right )\right \}\right \}\] ✓ Maple : cpu = 0.099 (sec), leaf count = 152
dsolve({a*diff(x(t),t)+b*diff(y(t),t) = alpha*x(t)+beta*y(t), b*diff(x(t),t)-a*diff(y(t),t) = beta*x(t)-alpha*y(t)})
\[\left \{x \left (t \right ) = c_{1} {\mathrm e}^{\frac {\left (\left (i \beta +\alpha \right ) a -b \left (i \alpha -\beta \right )\right ) t}{a^{2}+b^{2}}}+c_{2} {\mathrm e}^{-\frac {t \left (\left (i \beta -\alpha \right ) a -b \left (i \alpha +\beta \right )\right )}{a^{2}+b^{2}}}, y \left (t \right ) = i \left (c_{1} {\mathrm e}^{\frac {\left (\left (i \beta +\alpha \right ) a -b \left (i \alpha -\beta \right )\right ) t}{a^{2}+b^{2}}}-c_{2} {\mathrm e}^{-\frac {t \left (\left (i \beta -\alpha \right ) a -b \left (i \alpha +\beta \right )\right )}{a^{2}+b^{2}}}\right )\right \}\]