\[ \left \{-t^2+x'(t)+y(t)+6 t+1=0,y'(t)-x(t)=-3 t^2+3 t+1\right \} \] ✓ Mathematica : cpu = 0.0820949 (sec), leaf count = 124
DSolve[{1 + 6*t - t^2 + y[t] + Derivative[1][x][t] == 0, -x[t] + Derivative[1][y][t] == 1 + 3*t - 3*t^2},{x[t], y[t]},t]
\[\left \{\left \{x(t)\to \cos (t) \left (\left (3 t^2-t-13\right ) \cos (t)+(t-12) t \sin (t)\right )-\sin (t) \left (\left (-3 t^2+t+13\right ) \sin (t)+(t-12) t \cos (t)\right )+c_1 \cos (t)-c_2 \sin (t),y(t)\to \cos (t) \left (\left (-3 t^2+t+13\right ) \sin (t)+(t-12) t \cos (t)\right )+\sin (t) \left (\left (3 t^2-t-13\right ) \cos (t)+(t-12) t \sin (t)\right )+c_2 \cos (t)+c_1 \sin (t)\right \}\right \}\] ✓ Maple : cpu = 0.044 (sec), leaf count = 42
dsolve({diff(y(t),t)-x(t) = -3*t^2+3*t+1, diff(x(t),t)+y(t)-t^2+6*t+1 = 0})
\[\{x \left (t \right ) = \sin \left (t \right ) c_{2}+\cos \left (t \right ) c_{1}+3 t^{2}-t -13, y \left (t \right ) = t^{2}-c_{2} \cos \left (t \right )+c_{1} \sin \left (t \right )-12 t\}\]