While working on a physics problem I came across the following system of differential equations $$ \dot x(t)=\frac{v(w t - x)}{\sqrt{w ^2 t^2 - 2 w t x + x^2+y^2}}, $$ $$ \dot y(t)=\frac{-v y}{\sqrt{w ^2 t^2 - 2 w t x + x^2+y^2}}, $$ where on the right hand side I omitted the explicit dependence of $x=x(t)$ and $y=y(t)$ for better readability. The boundary conditions I wanted to consider are $$ x(0)=0, $$ $$ y(0)=d. $$ Originally, I tried to solve this for arbitrary real (but constant) parameters $v$, $w$, and $d$. However, even in the special case $v=1$, $w=0$, and $d$ arbitrary I struggle to solve it analytically. In this case the system above reduces to $$ \dot x(t)=\frac{- x}{\sqrt{x^2+y^2}}, $$ $$ \dot y(t)=\frac{- y}{\sqrt{x^2+y^2}}, $$ with the boundary conditions unchanged.
I can find the numerical solution with Mathematica for specific real parameters for the general problem. I also have no problem in guessing the solution for the reduced problem (I can even guess that the point $(x(t)=0,y(t)=0)$ must be excluded in this case). Thus, I guess that the actual solution must be easy as well. However, I cannot find an elegant analytic solution for either case.
Does this system of differential equations even have a (unique) solution? If so, what is the approach I am missing? I am grateful for any tips.