eqns = {x'[t] ==
    x[t] - y[t] -
     x[t] (x[t]^2 + y[t]^2) + (x[t] y[t])/Sqrt[x[t]^2 + y[t]^2],
   y'[t] ==
    x[t] + y[t] - y[t] (x[t]^2 + y[t]^2) -
     x[t]^2/Sqrt[x[t]^2 + y[t]^2]};
DSolve[eqns, {x, y}, t]