For my PDE class I have to find the flow of the following vector field
$$\mathbf{F}: \mathbb{R}^2-\{0\} \to \mathbb{R}^2, \mathbf{F}(x_1,x_2) = \frac{1}{r}(-x_2, x_1)$$ where $$r = \frac{1}{\sqrt{x_1^2 + x_2^2}}$$
I know that I can find the flow of this vector field by setting
$$\mathbf{\dot{x}}(t) = \mathbf{F}(\mathbf{x}(t)) $$ which yields the equations
$$ \dot{x_1} = \frac{1}{r} (-x_2) $$
and $$ \dot{x_2} = \frac{1}{r} x_1 $$ or in matrix notation $$ \dot{\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}} = \begin{pmatrix} 0 & -\frac{1}{r} \\ \frac{1}{r} & 0 \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} $$
Normally, I would solve this quite easily with an exponential ansatz. But here, $r$ itself depends on $x$ and $y$ which makes the matrix nonlinear. Thus I have no idea how to approach this?
The hint is valuable: the integral curves are always orthogonal to the position vector: it points to the idea the curves are circles and the factor $1/r$ doesn't change this.
If we need only the
integral curvesorbits, we can calculate $\dfrac{\dot x_2}{\dot x_1}=\dfrac{d x_2}{d x_1}$$\dfrac{d x_2}{d x_1}=-\dfrac{x_1}{x_2}$
$x_2=\pm\sqrt{c_1-x_1^2}$
Corresponding to circles centered at $(0,0)$
Added
Anyway, it's interesting solve the system for $t$ as it explains the factor $1/r$ when compared with the solution for the system without this factor.