Not all mathematical solutions to physical equations are physically meaningful: e.g., if we reverse all the molecular velocities in a breaking cup, we get pieces self-assembling into a cup. The resulting initial conditions are "degenerate": once we modify them, self-assembly stops. So, in a physical solution, the initial conditions must be "non-degenerate".
A challenge in formalizing this idea is that it depends on the representation. Example 1: we can use the Schroedinger equation to represent the potential field V(x)=F(f,...) as a function of the wave function f(x,t) and its derivatives. The new equation dF/dt=0 is equivalent to the Schroedinger equation, but now V(x) is in the initial conditions.
Example 2: for a general scalar field f, we describe a new equation which is satisfied if f satisfies the Euler-Lagrange equations for some Lagrangian L. So, similarly to Wheeler's cosmological "mass without mass", we have "equations without equations".
Thus, when formalizing physical equations, we must not only describe them in a mathematical form, we must also select one of the mathematically equivalent forms.