Can an Equation of Motion Do More? My usual expectation is that an equation of motion should give me the time-evolution of a system given an initial condition. But I am curious as to can an equation of motion do more than that? In particular, can an equation of motion dictate something about the initial condition itself--for example, how much energy could an initial state have or what should be the spatial pattern followed by the initial state? 
Clearly, Schrodinger equation seems to do so: We can have a particle with only certain energies inside the infinite square well of a given length, and there is always the time-independent Schrodinger equation that gives us a spatial pattern that the wave-function should obey at a given instant of time (or, in other words, a spatial pattern that the initial condition should obey). 
Is it really astonishing that Schrodinger equation does more than just predicting time-evolution? Or is it a generic feature of all equations of motion that they confine more things than just the time-evolution? 
 A: 
In particular, can an equation of motion dictate something about the initial condition itself--for example, how much energy could an initial state have or what should be the spatial pattern followed by the initial state? 

Classical equations  of motion are solutions of simple differential equations .The only constaint comes from the potential introduced in the differential equation,   that can dictate initial conditions. Once integrated, one can impose initial conditions to the solutions to fit the data : the trajectory of a rocket in a gravitational field for example. Here is the differential equation for simple harmonic motion
The equation  has specific constraints in the value of the constants entering the equations, and  the solutions need initial conditions to be useful: predict real numbers. The solutions cannot dictate initial conditions, only accept them.

Schrodinger equation that gives us a spatial pattern that the wave-function should obey at a given instant of time . 

The Shrodinger equation is a general differential equation , it is  an analogue in different variables of an equations of motion in a potential, but the solutions of the differential equation  give probability distributions for the energy state of a specifically chosen system, not particle trajectories.

(or, in other words, a spatial pattern that the initial condition should obey)

The initial conditions (the potential chosen and the constants)  dictate a pattern for the wavefunction. The energy levels are a prediction of the equation, initial conditions, and form of solutions. The initial conditions allow for a large number of solutions: hydrogen atom, helium atom....
At most one can say that the initial conditions are chosen so as to fit the problem at hand, and that a specific differential form is interdependent to the specific  problem under solution, (hydrogen or helium). It all has to hang together. The general form though of the differential equations and its solutions accepts initial conditions, does not dictate them. The same is true for the classical equations of motion used for specific problems.
A: The expectation that an equation of motion should spit-out the time-evolution and not determine the initial condition in any non-trivial manner is, of course, correct. The expectation that the equation of motion should not dictate the spatial variation of the wave-function over and above simply carrying forward (in time) the spatial pattern dictated by the initial conditions is also correct. The apparent confusion with these expectations in regard to the Schrodinger equation arises purely out of an utter misunderstanding of the machinery of quantum mechanics and of the logic behind the time-independent Schrodinger equation.

We can have a particle with only certain energies inside the infinite square well of a given length [.]

This is not exactly correct. The correct thing to say is that we can have energy-eigenstates of the particle inside the infinite square well of only certain values of energy. The particle, in the initial state (and thus, in the subsequent states) can obviously be in a superposition of a number of such eigenstates and thus, the time-independent Schrodinger equation, via determining the energy eigenvalues and eigenstates, doesn't determine the initial condition at all. 
Now, it seems nonetheless interesting as to why the equation of motion would determine even the energy eigenvalues. The answer lies in the fact that the time-independent Schrodinger equation is not really the equation of motion. It is the equation followed by the normal modes of the full time-dependent Schrodinger equation (which is the actual equation of motion) and the fact that it determines the energy eigenvalues is a direct consequence of the fact that the normal modes of the full time-dependent Schrodinger equation are also the eigenstates of the Hamiltonian because the Hamiltonian is the generator of translations in time. This is again no scary coincidence that energy eigenstates happen to be the eigenstates of the operator that generates translations in time, because, thanks to Emmy Noether, we understand that energy literally means the eigenvalues of the operator that generates translations in time. With this understanding, the next confusion evaporates in the heat of its own shame but nonetheless, I will briefly address it.  

[T]he time-independent Schrodinger equation [...] gives us a spatial
  pattern that the wave-function should obey at a given instant of time
  (or, in other words, a spatial pattern that the initial condition
  should obey).

As expressed earlier, the time-independent Schrodinger equation is simply the eigenvalue equation of the Hamiltonian. Thus, it doesn't determine the spatial pattern of a generic initial condition but it simply determines the spatial pattern of an energy eigenstate which, for the reason that Hamiltonian is the generator of translations in time, would also be a stationary state (or, a normal mode) of the full time-dependent Schrodinger equation. Thus, all it says is that if a state wants to be the energy eigenstate then it has to have a spatial profile that follows the time-independent Schrodinger equation. But, an initial state clearly doesn't need to want to be an energy eigenstate. 
