\[ \left \{t^2 (1-\sin (t)) x'(t)=t^2 y(t)+t x(t) (1-2 \sin (t)),t^2 (1-\sin (t)) y'(t)=x(t) (t \cos (t)-\sin (t))+t y(t) (1-t \cos (t))\right \} \] ✓ Mathematica : cpu = 0.0158174 (sec), leaf count = 29
DSolve[{t^2*(1 - Sin[t])*Derivative[1][x][t] == t*(1 - 2*Sin[t])*x[t] + t^2*y[t], t^2*(1 - Sin[t])*Derivative[1][y][t] == (t*Cos[t] - Sin[t])*x[t] + t*(1 - t*Cos[t])*y[t]},{x[t], y[t]},t]
\[\left \{\left \{x(t)\to c_1 t^2+c_2 t,y(t)\to c_1 t+c_2 \sin (t)\right \}\right \}\] ✓ Maple : cpu = 0.091 (sec), leaf count = 23
dsolve({t^2*(1-sin(t))*diff(x(t),t) = t*(1-2*sin(t))*x(t)+t^2*y(t), t^2*(1-sin(t))*diff(y(t),t) = (t*cos(t)-sin(t))*x(t)+t*(1-t*cos(t))*y(t)})
\[\{x \left (t \right ) = t \left (t c_{1}+c_{2}\right ), y \left (t \right ) = \sin \left (t \right ) c_{2}+t c_{1}\}\]