\[ \left \{x''(t)+6 x(t)+7 y(t)=0,3 x(t)+y''(t)+2 y(t)=2 t\right \} \] ✓ Mathematica : cpu = 0.871432 (sec), leaf count = 742
DSolve[{6*x[t] + 7*y[t] + Derivative[2][x][t] == 0, 3*x[t] + 2*y[t] + Derivative[2][y][t] == 2*t},{x[t], y[t]},t]
\[\left \{\left \{x(t)\to -\frac {7}{200} e^{-t} \left (e^{2 t}-2 e^t \cos (3 t)+1\right ) \left (-7 e^{-t} \left (e^{2 t} (t-1)+t+1\right )-\frac {2}{9} \sin (3 t)+\frac {2}{3} t \cos (3 t)\right )+\frac {7}{600} e^{-t} \left (3 e^{2 t}+14 e^t \cos (3 t)+3\right ) \left (3 e^{-t} \left (e^{2 t} (t-1)+t+1\right )-\frac {2}{9} \sin (3 t)+\frac {2}{3} t \cos (3 t)\right )-\frac {7}{600} e^{-t} \left (9 e^{2 t}+14 e^t \sin (3 t)-9\right ) \left (e^{-t} \left (e^{2 t} (t-1)-t-1\right )-\frac {2}{3} t \sin (3 t)-\frac {2}{9} \cos (3 t)\right )-\frac {7}{600} e^{-t} \left (3 e^{2 t}-2 e^t \sin (3 t)-3\right ) \left (7 e^t (t-1)-7 e^{-t} (t+1)+2 t \sin (3 t)+\frac {2}{3} \cos (3 t)\right )-\frac {7}{20} c_3 e^{-t} \left (e^{2 t}-2 e^t \cos (3 t)+1\right )+\frac {1}{20} c_1 e^{-t} \left (3 e^{2 t}+14 e^t \cos (3 t)+3\right )-\frac {7}{60} c_4 e^{-t} \left (3 e^{2 t}-2 e^t \sin (3 t)-3\right )+\frac {1}{60} c_2 e^{-t} \left (9 e^{2 t}+14 e^t \sin (3 t)-9\right ),y(t)\to -\frac {7}{200} e^{-t} \left (e^{2 t}-2 e^t \cos (3 t)+1\right ) \left (3 e^{-t} \left (e^{2 t} (t-1)+t+1\right )-\frac {2}{9} \sin (3 t)+\frac {2}{3} t \cos (3 t)\right )+\frac {1}{200} e^{-t} \left (7 e^{2 t}+6 e^t \cos (3 t)+7\right ) \left (-7 e^{-t} \left (e^{2 t} (t-1)+t+1\right )-\frac {2}{9} \sin (3 t)+\frac {2}{3} t \cos (3 t)\right )+\frac {7}{200} e^{-t} \left (3 e^{2 t}-2 e^t \sin (3 t)-3\right ) \left (e^{-t} \left (e^{2 t} (t-1)-t-1\right )-\frac {2}{3} t \sin (3 t)-\frac {2}{9} \cos (3 t)\right )+\frac {1}{200} e^{-t} \left (7 e^{2 t}+2 e^t \sin (3 t)-7\right ) \left (7 e^t (t-1)-7 e^{-t} (t+1)+2 t \sin (3 t)+\frac {2}{3} \cos (3 t)\right )-\frac {3}{20} c_1 e^{-t} \left (e^{2 t}-2 e^t \cos (3 t)+1\right )+\frac {1}{20} c_3 e^{-t} \left (7 e^{2 t}+6 e^t \cos (3 t)+7\right )-\frac {1}{20} c_2 e^{-t} \left (3 e^{2 t}-2 e^t \sin (3 t)-3\right )+\frac {1}{20} c_4 e^{-t} \left (7 e^{2 t}+2 e^t \sin (3 t)-7\right )\right \}\right \}\] ✓ Maple : cpu = 0.061 (sec), leaf count = 64
dsolve({diff(diff(x(t),t),t)+6*x(t)+7*y(t) = 0, diff(diff(y(t),t),t)+3*x(t)+2*y(t) = 2*t})
\[\left \{x \left (t \right ) = \frac {14 t}{9}+c_{1} {\mathrm e}^{t}+c_{2} \cos \left (3 t \right )+c_{3} {\mathrm e}^{-t}+c_{4} \sin \left (3 t \right ), y \left (t \right ) = -c_{1} {\mathrm e}^{t}+\frac {3 c_{2} \cos \left (3 t \right )}{7}-c_{3} {\mathrm e}^{-t}+\frac {3 c_{4} \sin \left (3 t \right )}{7}-\frac {4 t}{3}\right \}\]