Considering an ordinary differential equation of first order in the implicit form
$$ F(q(t),\dot q (t))=\alpha,\,\,\, q(0)=q_0 $$
with $\alpha,\, q_0$ constants, what is the relation of the solution $q(t)$ to solutions of the variational problem for $S[y]$, obtained by taking $F$ (or a suitable function thereof, e.g. $F^2$) as a Lagrange function
$$ S[y] = \int_0^T F(y(t),\dot y (t))\, dt, \\ y(0)=q_0, \, \, \, y(T)=q(T) $$
where $T$ is a conveniently chosen positive constant.
$\bf{Corrected:}$
The solutions of the variational problem for $F(y(t),\dot y (t))$ imply another implicit equation
$$F(y(t),\dot y (t))-\dot y(t) \frac{\partial F}{\partial {\dot y(t)}} = \it{const.}$$
With $\dot y(t) = z$, the correct function $G(y,z)$ to take in the variational problem should therefore satify
$$ \frac{\partial}{\partial z} \left( \frac{G}{z} \right) =\frac{F(y,z)}{z^2} $$
which relates $G$ to $F$ as a Lagrangian relates to its corresponding Hamilton function.
It seems therefore that a variational problem can be associated to any implicit, first order ODE.