\[ \left \{t x'(t)+y(t)=0,x(t)+t y'(t)=0\right \} \] ✓ Mathematica : cpu = 0.0040598 (sec), leaf count = 31
DSolve[{y[t] + t*Derivative[1][x][t] == 0, x[t] + t*Derivative[1][y][t] == 0},{x[t], y[t]},t]
\[\left \{\left \{x(t)\to c_1 t+\frac {c_2}{t},y(t)\to \frac {c_2}{t}-c_1 t\right \}\right \}\] ✓ Maple : cpu = 0.045 (sec), leaf count = 31
dsolve({t*diff(x(t),t)+y(t) = 0, t*diff(y(t),t)+x(t) = 0})
\[\left \{x \left (t \right ) = \frac {t^{2} c_{2}+c_{1}}{t}, y \left (t \right ) = \frac {-t^{2} c_{2}+c_{1}}{t}\right \}\]