For what it worth, Euler-Lagrange (EL) equations and Newton's 2nd law themselves are just differential equations, nothing more. The context provides appropriate conditions, such as, e.g., initial value conditions or boundary value conditions.
See also this & this related Phys.SE posts and this related Math.SE post.
Unlike what OP seems to suggest (v3), Newton's 2nd law and Euler-Lagrange (EL) equations are strictly speaking just differential equations (DEs) without conditions. Rather the context provides the appropriate conditions, such as, e.g., initial conditions (ICs) or boundary conditions (BCs). Together with the DEs, they constitute an initial value problem (IVP) or a boundary value problem (BVP), respectively.
The issues of ICs vs. BCs for the principle of stationary action are already covered in this & this related Phys.SE posts and this related Math.SE post.