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.
Since the Schrödinger equation is linear it does not fix the norm of its solutions. You have to do that manually at your desk.