7.3: Behaviour for large |y|

Before solving the equation we are going to see how the solutions behave at large $|y|$ (and also large $|x|$, since these variable are proportional!). For $|y|$ very large, whatever the value of $\epsilon, \epsilon \ll y^2$, and thus we have to solve

$$\frac{d^2 u}{d y^2}=y^2 u(y)$$

This has two type of solutions, one proportional to $e^{y^2 / 2}$ and one to $e^{-y^2 / 2}$. We reject the first one as being not normalisable.
Question: Check that these are the solutions. Why doesn't it matter that they don't exactly solve the equations?
Substitute $u(y)=H(y) e^{-y^2 / 2}$. We find

$$\frac{d^2 u}{d y^2}=\left [H^{\prime \prime}(y)-2 y H^{\prime}(y)+y^2 H(y)] e^{-y^2/ 2}\right ].$$

so we can obtain a differential equation for $H(y)$ in the form

$$H^{\prime \prime}(y)-2 y H^{\prime}(y)+(2 \epsilon-1) H(y)=0 .$$

This equation will be solved by a substitution and infinite series (Taylor series!), and showing that it will have to terminates somewhere, i.e., $H(y)$ is a polynomial!