Quantum Harmonic Oscillator- Solving the Differential Equation at a Limit The eigenvalue equation for the quantum harmonic oscillator is $$\langle y | E\rangle '' +(2\epsilon-y^2)\langle y| E \rangle=0$$ where $\epsilon = \frac{E}{\hbar\omega}$ and $y=\sqrt{\frac{\hbar}{m\omega}}x$. In Shankar's book, he starts to solve this by taking the limit at infinity, making the equation $$\langle y | E\rangle '' -y^2\langle y| E \rangle=0$$
Apparently, the solution to this equation in the same limit is $\langle y | E\rangle = Ay^me^{\pm\frac{1}{2}y^2}$. I know how to work with kind of equation normally, but how do I solve this equation in the limit $ y \to \infty$? 
 A: Define $\langle y | E\rangle = \psi_(y)$ and study 
the asymptotics of your ODE,
$$
(\partial_y^2-y^2)~\psi (y)=0.
$$
That means the leading behavior of ψ for large y;  so , for example, if you had a polynomial in y, you just keep the highest order thereof: If you have an order-m polynomial, you'd just keep the $y^m$ monomial, since it dominates all lower powers in y at large y. 
So, does  $Ay^me^{\pm\frac{1}{2}y^2}$ satisfy your differential equation to leading order in y? Does $$\partial_y^2 (y^me^{\pm\frac{1}{2}y^2}) = (y^{m+2}+ O(y^{m+1} )~   ) e^{\pm\frac{1}{2}y^2}   ~~?$$ Let's see:
$$
\partial_y^2 (y^me^{\pm\frac{1}{2}y^2}) = \partial_y \Bigl ((m/y\pm y) y^me^{\pm\frac{1}{2}y^2}\Bigr )\\ = ((m/y\pm y)^2-m/y^2\pm 1)~y^m e^{\pm\frac{1}{2}y^2}\\
= (y^2 \pm 2m + m^2/y^2 -m/y^2 \pm 1  )~~y^m  e^{\pm\frac{1}{2}y^2},
$$
So, indeed,  the leading behavior of the parenthesis on the r.h.s. is $y^2$, your desideratum. 

When you solve the full equation at all y, beyond asymptotics,
$$
(\partial_y^2- y^2 + 2m +1)\psi_m(y)=0,
$$
and you dismiss the solutions blowing up for large y, you find a celebrated  solution, the Hermite functions,
$$
\psi_m(y)\propto (-)^m e^{y^2/2} ~\partial_y^m e^{-y^2} =  e^{-y^2/2} \Bigl (2y -\partial_y   \Bigr)^m \cdot 1 ,
$$
which, presumably, your text is building up motivation for. They are the eigenfunctions of the Fourier transform.
