\[ \left \{x'(t)+2 x(t)+y'(t)+y(t)=t+e^{2 t},x'(t)-x(t)+y'(t)+3 y(t)=e^t-1\right \} \] ✓ Mathematica : cpu = 0.0911166 (sec), leaf count = 118
DSolve[{2*x[t] + y[t] + Derivative[1][x][t] + Derivative[1][y][t] == E^(2*t) + t, -x[t] + 3*y[t] + Derivative[1][x][t] + Derivative[1][y][t] == -1 + E^t},{x[t], y[t]},t]
\[\left \{\left \{x(t)\to \frac {1}{5} \left (t-e^t+e^{2 t}+1\right )+\frac {5}{72} \left (\frac {12 \left (5712 t+833 e^t+2352 e^{2 t}-5508\right )}{20825}+c_1 e^{-7 t/5}\right ),y(t)\to \frac {1}{5} \left (-t+e^t-e^{2 t}-1\right )+\frac {5}{48} \left (\frac {12 \left (5712 t+833 e^t+2352 e^{2 t}-5508\right )}{20825}+c_1 e^{-7 t/5}\right )\right \}\right \}\] ✓ Maple : cpu = 0.071 (sec), leaf count = 51
dsolve({diff(x(t),t)+diff(y(t),t)-x(t)+3*y(t) = exp(t)-1, diff(x(t),t)+diff(y(t),t)+2*x(t)+y(t) = exp(2*t)+t})
\[\left \{x \left (t \right ) = \frac {3 t}{7}-\frac {1}{49}-\frac {{\mathrm e}^{t}}{6}+\frac {5 \,{\mathrm e}^{2 t}}{17}+{\mathrm e}^{-\frac {7 t}{5}} c_{1}, y \left (t \right ) = -\frac {{\mathrm e}^{2 t}}{17}+\frac {t}{7}-\frac {26}{49}+\frac {{\mathrm e}^{t}}{4}+\frac {3 \,{\mathrm e}^{-\frac {7 t}{5}} c_{1}}{2}\right \}\]