\[ \left \{\left (-t^2+x(t)^2+y(t)^2\right ) x'(t)=-2 t x(t),\left (-t^2+x(t)^2+y(t)^2\right ) y'(t)=-2 t y(t)\right \} \] ✓ Mathematica : cpu = 0.0661912 (sec), leaf count = 191
DSolve[{(-t^2 + x[t]^2 + y[t]^2)*Derivative[1][x][t] == -2*t*x[t], (-t^2 + x[t]^2 + y[t]^2)*Derivative[1][y][t] == -2*t*y[t]},{x[t], y[t]},t]
\[\left \{\left \{y(t)\to \frac {c_1 \left (e^{c_2}-\sqrt {-4 t^2-4 c_1{}^2 t^2+e^{2 c_2}}\right )}{2 \left (1+c_1{}^2\right )},x(t)\to \frac {e^{c_2}-\sqrt {-4 t^2-4 c_1{}^2 t^2+e^{2 c_2}}}{2 \left (1+c_1{}^2\right )}\right \},\left \{y(t)\to \frac {c_1 \left (\sqrt {-4 t^2-4 c_1{}^2 t^2+e^{2 c_2}}+e^{c_2}\right )}{2 \left (1+c_1{}^2\right )},x(t)\to \frac {\sqrt {-4 t^2-4 c_1{}^2 t^2+e^{2 c_2}}+e^{c_2}}{2 \left (1+c_1{}^2\right )}\right \}\right \}\] ✓ Maple : cpu = 0.319 (sec), leaf count = 180
dsolve({(x(t)^2+y(t)^2-t^2)*diff(x(t),t) = -2*t*x(t), (x(t)^2+y(t)^2-t^2)*diff(y(t),t) = -2*t*y(t)})
\[\left [\{x \left (t \right ) = 0\}, \left \{y \left (t \right ) = \frac {1+\sqrt {-4 t^{2} c_{1}^{2}+1}}{2 c_{1}}, y \left (t \right ) = \frac {1-\sqrt {-4 t^{2} c_{1}^{2}+1}}{2 c_{1}}\right \}\right ]\]