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The Beltrami Identity:
If the Lagrangian $\:L\plr{y,y',x}\:$ of a system does not depend explicitly on $\:x$, that is \begin{equation} \dfrac{\partial L}{\partial x}\e 0 \tl{01} \end{equation} then from the Euler-Lagrange equation \begin{equation} \dfrac{\mr d}{\mr dx}\plr{\dfrac{\partial L}{\partial y'}}\m\dfrac{\partial L}{\partial y}\e 0 \tl{02} \end{equation} we have \begin{equation} \dfrac{\mr d}{\mr dx}\plr{y'\dfrac{\partial L}{\partial y'}\m L}\e 0 \tl{03} \end{equation} so \begin{equation} \boxed{\:\:y'\dfrac{\partial L}{\partial y'}\m L\e \texttt{constant}\:\:}\quad \texttt{(Beltrami Identity)} \tl{04} \end{equation}
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For your Lagrangian \begin{equation} \begin{split} \frac{\sqrt{\dot x ^2 \p \dot y ^2}}{y}\mr dt & \e \frac{\sqrt{1\p\plr{\dfrac{\dot y}{\dot x}}^2}}{y}\dot x\,\mr dt\e\frac{\sqrt{1\p\plr{\dfrac{\mr dy/\mr dt}{\mr dx/\mr dt}}^2}}{y}\dfrac{\mr dx}{\mr dt}\,\mr dt\\ &\e\frac{\sqrt{1\p\plr{\dfrac{\mr dy}{\mr dx}}^2}}{y}\mr dx\e\frac{\sqrt{1\p y'^{2}}}{y}\mr dx\\ \end{split} \tl{05} \end{equation} that is \begin{equation} L\plr{y,y',x}\e\frac{\sqrt{1\p y'^{2}}}{y} \tl{06} \end{equation}
Using the Lagrangian \eqref{06} we could find the $\:x\m$parametric representation $\:\blr{x,y\plr{x}}\:$ of the curve directly bypassing its $\:t\m$parametric representation $\:\blr{x\plr{t},y\plr{t}}$, that is the equations of the motion.
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Hint for the Solution
Insert the Lagrangian \eqref{06} in the Beltrami Identity \eqref{04} to find \begin{equation} f\plr{y,y'\e\dfrac{\mr dy}{\mr dx}}\e a\e \texttt{positive constant} \tl{H-01} \end{equation} Solve equation \eqref{H-01} with respect to $\:\mr dx$ to find \begin{equation} \mr dx\e g\plr{y}\mr dy \tl{H-02} \end{equation} In equation \eqref{H-02} make a proper convenient change from the variable $\:y\:$ to an angle variable $\:\theta\:$ \begin{equation} y\e h\plr{\theta} \tl{H-03} \end{equation} Convert equation \eqref{H-02} to something like that \begin{equation} \mr dx\e q\plr{\theta}\mr d\theta \tl{H-04} \end{equation} Integrate equation \eqref{H-04} to have \begin{equation} x\e u\plr{\theta} \tl{H-05} \end{equation} Equations \eqref{H-03} and \eqref{H-05} give a $\:\theta\m$parametric representation of the motion orbit.