\[ \left \{x'(t)+y'(t)-y(t)=e^t,2 x'(t)+y'(t)+2 y(t)=\cos (t)\right \} \] ✓ Mathematica : cpu = 0.133942 (sec), leaf count = 122
DSolve[{-y[t] + Derivative[1][x][t] + Derivative[1][y][t] == E^t, 2*y[t] + 2*Derivative[1][x][t] + Derivative[1][y][t] == Cos[t]},{x[t], y[t]},t]
\[\left \{\left \{x(t)\to -\frac {3}{4} c_2 \left (e^{4 t}-1\right )+\frac {1}{68} e^{-4 t} \left (e^{4 t}-1\right ) \left (34 e^t+3 \sin (t)-12 \cos (t)\right )+\frac {1}{4} \left (2 e^{-3 t}+2 e^t+\frac {3}{17} e^{-4 t} \sin (t)+\sin (t)-\frac {12}{17} e^{-4 t} \cos (t)\right )+c_1,y(t)\to \frac {1}{51} \left (-34 e^t-3 \sin (t)+12 \cos (t)\right )+c_2 e^{4 t}\right \}\right \}\] ✓ Maple : cpu = 0.13 (sec), leaf count = 47
dsolve({diff(x(t),t)+diff(y(t),t)-y(t) = exp(t), 2*diff(x(t),t)+diff(y(t),t)+2*y(t) = cos(t)})
\[\left \{x \left (t \right ) = \frac {{\mathrm e}^{4 t} c_{1}}{4}+\frac {5 \sin \left (t \right )}{17}-\frac {3 \cos \left (t \right )}{17}+{\mathrm e}^{t}+c_{2}, y \left (t \right ) = -\frac {{\mathrm e}^{4 t} c_{1}}{3}+\frac {4 \cos \left (t \right )}{17}-\frac {\sin \left (t \right )}{17}-\frac {2 \,{\mathrm e}^{t}}{3}\right \}\]