How to solve $\frac{dx}{ds}=\frac{x}{x^2+y^2},\quad\frac{dy}{ds}=\frac{y}{x^2+y^2}$? I was wondering if my method to solve the following pair of equations:
$$\frac{dx}{ds}=\frac{x}{x^2+y^2},\quad\frac{dy}{ds}=\frac{y}{x^2+y^2}$$
is the right one.
My attempt: We have $\frac{x^2+y^2}{x}dx=ds$ and $\frac{x^2+y^2}{y}dy=ds$, so $$\frac{1}{x}dx=\frac{1}{y}dy.$$
The general solution of the first equation is: $y=Ax$, where $A$ is a constant.
Thank you in advance.

You have $$\frac{\text{d}}{\text{d}s}\,\left(x^2+y^2\right)=2\,.$$
So, $x^2+y^2=2(s+c)$ for some constant $c$.  Plugging this in, you will see that $$x=\pm\sqrt{2A(s+c)}\text{ and }y=\pm\sqrt{2(1-A)(s+c)}$$ for some constant $A$.
