Ansätze to solve 2-dimensional nonlinear integral curve I try to solve the initial value problem analytically $X(t_0) = X_0, X \in \mathbb{R}^2$ for the following integral curve:
$x'(t) = -y(t)+x(t) (x(t)^2+y(t)^2-1) \\
y'(t) = x(t)+y(t) (x(t)^2+y(t)^2-1) $
According to the Picard–Lindelöf theorem there should be a solution for at least some period of time since the system is locally Lipschitz continuous. All differential equations I solved analytically so far were either one dimensional (seperation of variables, using $e^\lambda$) or linear (variation of parameters).
So far I figured out, that trajectories will converge towards the unit circle. But I still can not figure out a method how to calcluate an analytical solution. 
Is there a known method to solve these systems? I would be gratefull just for the hint to a method I could use. Or is it completely hopeless to find a closed solution for this problem.
 A: Try the following. We expand both equations by $y$ (first equation) and by $x$ (second equation).
$$x'y=-y^2+xy(x^2+y^2-1)$$
$$y'x=x^2+xy(x^2+y^2-1)$$
Now subtract both equations
$$y'x-x'y=x^2+y^2$$
After deviding by $x^2$, we obtain:
$$\frac{y'x-yx'}{x^2}=1+\frac{y^2}{x^2}$$
Notice that we have the derivative of $y/x$ on the left side:
$$\frac{d}{dt}\left(\frac{y}{x}\right)=1+\frac{y^2}{x^2}$$
Substitute $u(t)=\frac{y}{x}$ to get 
$$\frac{d u}{dt}=1+u^2$$
This DE has $u(t)=\tan(t)$ as solution.
Now you can substitute $y(t)=x(t)\tan(t)$ into the second equation. Which will be in $x$ and $t$ only.
A: A big hint is that, if a solution $(x_c(t),y_c(t))$ lies on the unit circle, then you have $x_c(t)^2 + y_c(t)^2 = 1$. Assume that there exists a solution to your system which entirely lies on the unit circle, so assume that $(x_c(t),y_c(t))$ is a solution to your dynamical system. Then, as you can quickly see, for a solution on the unit circle, your system simplifies quite a lot. In particular, you can solve the resulting system for $x_c(t),y_c(t))$ quite easily, and obtain an explicit expression for the solution of your dynamical system on the unit circle.
Alternatively, you can try to write the system in polar coordinates. That is, you transform your cartesian coordinates $(x,y)$ to
\begin{align}
x(t) &= r(t) \cos \theta(t),\\
y(t) &= r(t) \sin \theta(t),
\end{align}
with $r(t) > 0$. This also gives more insight into the dynamics of the system for solutions which do not lie on the unit circle, if you're interested.
