\[ \left \{x''(t)+x(t)+y(t)=-5,-4 x(t)+y''(t)-3 y(t)=-3\right \} \] ✓ Mathematica : cpu = 0.197421 (sec), leaf count = 554
DSolve[{x[t] + y[t] + Derivative[2][x][t] == -5, -4*x[t] - 3*y[t] + Derivative[2][y][t] == -3},{x[t], y[t]},t]
\[\left \{\left \{x(t)\to -\frac {1}{8} e^{-t} \left (e^{-t} (-13 t-10)+e^t (10-13 t)\right ) \left (e^{2 t} t+t-e^{2 t}+1\right )-\frac {1}{8} e^{-t} \left (e^{2 t}-1\right ) t \left (e^{-t} (-13 t-23)+e^t (13 t-23)\right )-\frac {1}{8} e^{-t} \left (e^{2 t} t+t-2 e^{2 t}+2\right ) \left (e^t (13 t-23)+e^{-t} (13 t+23)\right )-\frac {1}{8} e^{-t} \left (e^{2 t} t-t-e^{2 t}-1\right ) \left (e^t (36-13 t)+e^{-t} (13 t+36)\right )-\frac {1}{4} c_4 e^{-t} \left (e^{2 t} t+t-e^{2 t}+1\right )-\frac {1}{2} c_1 e^{-t} \left (e^{2 t} t-t-e^{2 t}-1\right )-\frac {1}{2} c_2 e^{-t} \left (e^{2 t} t+t-2 e^{2 t}+2\right )-\frac {1}{4} c_3 e^{-t} \left (e^{2 t}-1\right ) t,y(t)\to \frac {1}{4} e^{-t} \left (e^{2 t}+1\right ) \left (e^{-t} (-13 t-10)+e^t (10-13 t)\right ) t+\frac {1}{4} e^{-t} \left (e^{2 t}-1\right ) \left (e^t (36-13 t)+e^{-t} (13 t+36)\right ) t+\frac {1}{4} e^{-t} \left (e^{2 t} t-t+e^{2 t}+1\right ) \left (e^{-t} (-13 t-23)+e^t (13 t-23)\right )+\frac {1}{4} e^{-t} \left (e^{2 t} t+t-e^{2 t}+1\right ) \left (e^t (13 t-23)+e^{-t} (13 t+23)\right )+c_1 e^{-t} \left (e^{2 t}-1\right ) t+\frac {1}{2} c_4 e^{-t} \left (e^{2 t}+1\right ) t+c_2 e^{-t} \left (e^{2 t} t+t-e^{2 t}+1\right )+\frac {1}{2} c_3 e^{-t} \left (e^{2 t} t-t+e^{2 t}+1\right )\right \}\right \}\] ✓ Maple : cpu = 0.056 (sec), leaf count = 60
dsolve({diff(diff(x(t),t),t)+x(t)+y(t) = -5, diff(diff(y(t),t),t)-4*x(t)-3*y(t) = -3})
\[\{x \left (t \right ) = \left (t c_{4}+c_{2}\right ) {\mathrm e}^{-t}+18+\left (t c_{3}+c_{1}\right ) {\mathrm e}^{t}, y \left (t \right ) = \left (\left (-2 t +2\right ) c_{4}-2 c_{2}\right ) {\mathrm e}^{-t}-23+\left (\left (-2 t -2\right ) c_{3}-2 c_{1}\right ) {\mathrm e}^{t}\}\]