\[ \left \{x'(t)=-y(t) \left (x(t)^2+y(t)^2\right ),y'(t)=\left (\begin {array}{cc} \{ & \begin {array}{cc} x(t)^2+y(t)^2 & x(t)^2+y(t)^2\geq 2 x(t) \\ \left (x(t)^2+y(t)^2\right ) \left (\frac {x(t)}{2}-\frac {y(t)^2}{2 x(t)}\right ) & \text {True} \\\end {array} \\\end {array}\right )\right \} \] ✗ Mathematica : cpu = 1.98532 (sec), leaf count = 0
DSolve[{Derivative[1][x][t] == -(y[t]*(x[t]^2 + y[t]^2)), Derivative[1][y][t] == Piecewise[{{x[t]^2 + y[t]^2, x[t]^2 + y[t]^2 >= 2*x[t]}}, (x[t]^2 + y[t]^2)*(x[t]/2 - y[t]^2/(2*x[t]))]},{x[t], y[t]},t]
, could not solve
DSolve[{Derivative[1][x][t] == -(y[t]*(x[t]^2 + y[t]^2)), Derivative[1][y][t] == Piecewise[{{x[t]^2 + y[t]^2, x[t]^2 + y[t]^2 >= 2*x[t]}}, (x[t]^2 + y[t]^2)*(x[t]/2 - y[t]^2/(2*x[t]))]}, {x[t], y[t]}, t]
✗ Maple : cpu = 0. (sec), leaf count = 0
dsolve({diff(x(t),t) = -y(t)*(x(t)^2+y(t)^2), diff(y(t),t) = piecewise(2*x(t) <= x(t)^2+y(t)^2,x(t)^2+y(t)^2,(1/2*x(t)-1/2*y(t)^2/x(t))*(x(t)^2+y(t)^2))})
, could not solve
dsolve({diff(x(t),t) = -y(t)*(x(t)^2+y(t)^2), diff(y(t),t) = piecewise(2*x(t) <= x(t)^2+y(t)^2,x(t)^2+y(t)^2,(1/2*x(t)-1/2*y(t)^2/x(t))*(x(t)^2+y(t)^2))})