\[ \left \{a y(t)+x''(t)=0,y''(t)-a^2 y(t)=0\right \} \] ✓ Mathematica : cpu = 0.0144492 (sec), leaf count = 115
DSolve[{a*y[t] + Derivative[2][x][t] == 0, -(a^2*y[t]) + Derivative[2][y][t] == 0},{x[t], y[t]},t]
\[\left \{\left \{x(t)\to -\frac {c_4 e^{-a t} \left (-2 a t e^{a t}+e^{2 a t}-1\right )}{2 a^2}-\frac {c_3 e^{-a t} \left (e^{a t}-1\right )^2}{2 a}+c_2 t+c_1,y(t)\to \frac {1}{2} c_3 e^{-a t} \left (e^{2 a t}+1\right )+\frac {c_4 e^{-a t} \left (e^{2 a t}-1\right )}{2 a}\right \}\right \}\] ✓ Maple : cpu = 0.074 (sec), leaf count = 49
dsolve({diff(diff(x(t),t),t)+a*y(t) = 0, diff(diff(y(t),t),t)-a^2*y(t) = 0})
\[\left \{x \left (t \right ) = \frac {-c_{4} {\mathrm e}^{-a t}-c_{3} {\mathrm e}^{a t}+a \left (t c_{1}+c_{2}\right )}{a}, y \left (t \right ) = c_{3} {\mathrm e}^{a t}+c_{4} {\mathrm e}^{-a t}\right \}\]