\[ \left \{x'(t)-y(t)+z(t)=0,-x(t)+y'(t)-y(t)=t,-x(t)+z'(t)-z(t)=t\right \} \] ✓ Mathematica : cpu = 0.0162036 (sec), leaf count = 226
DSolve[{-y[t] + z[t] + Derivative[1][x][t] == 0, -x[t] - y[t] + Derivative[1][y][t] == t, -x[t] - z[t] + Derivative[1][z][t] == t},{x[t], y[t], z[t]},t]
\[\left \{\left \{x(t)\to e^{-t} \left (1-e^t\right ) (-t-1)+e^{-t} \left (e^t-1\right ) (-t-1)+c_2 \left (e^t-1\right )+c_3 \left (1-e^t\right )+c_1,y(t)\to e^{-t} (-t-1) \left (-e^t t+e^t-1\right )+e^{-t} (-t-1) \left (e^t t+1\right )+c_3 \left (-e^t t+e^t-1\right )+c_1 \left (e^t-1\right )+c_2 \left (e^t t+1\right ),z(t)\to e^{-t} (-t-1) \left (-e^t t+2 e^t-1\right )+e^{-t} (-t-1) \left (e^t t-e^t+1\right )+c_3 \left (-e^t t+2 e^t-1\right )+c_1 \left (e^t-1\right )+c_2 \left (e^t t-e^t+1\right )\right \}\right \}\] ✓ Maple : cpu = 0.062 (sec), leaf count = 51
dsolve({diff(x(t),t)-y(t)+z(t) = 0, diff(y(t),t)-x(t)-y(t) = t, diff(z(t),t)-x(t)-z(t) = t})
\[\{x \left (t \right ) = c_{2}+c_{3} {\mathrm e}^{t}, y \left (t \right ) = \left (t c_{3}+c_{1}\right ) {\mathrm e}^{t}-t -c_{2}-1, z \left (t \right ) = \left (\left (t -1\right ) c_{3}+c_{1}\right ) {\mathrm e}^{t}-t -c_{2}-1\}\]