Question: Why do some solutions of the Schrodinger equation emerge un-normalized? Logically, any solution psi must have its integral $\int\_{-\infty}^\infty |\varphi|^2\, dx = 1$ because the probability has to certain across all x. Hence any solution that comes out of solving the SWE must be normalized. Where am I going wrong?

Answer: 
> 
> Logically, any solution $\psi$ must have its integral
>  $\int\_{-\infty}^\infty|\phi|^2dx=1$ because the probability has to
>  certain across all x
> 
> 
> 

This simply isn't true for the reason that not all solutions to Schrodinger's equation (SE) correspond to a physical state. Put another way, only normalizable solutions to SE correspond to physical states.

For example, there are an uncountable infinity of solutions to the quantum harmonic oscillator SE *but* there are only a countable infinity of solutions that are normalizable and thus are representations of physical states.

It *is* logical that any solution $\psi$ to the SE that is a representation of a physical state is normalizable but it certainly *is not* logical that any solution to a partial differential equation such as the SE is normalizable.
What solutions are you referring to? If you're referring to solutions of the *time-independent Schrodinger equation*, e.g. the eigenfunctions, then these can be normalizable or not depending on your potential $V$. However, in general these solutions have *nothing to do* with the wavefunction of the actual particle. The wavefunction of an *actual particle* is always normalized, as you say. But the eigenfunctions don't need to be normalizable, because the eigenfunctions don't necessarily represent a physical state of a particle. They simply represent a convenient basis for expanding physical wavefunctions.

For more about this distinction, see the section in Griffiths, chapter 2, where he discusses constructing the wavefunction of a free particle.