How do I use Leibniz formula to solve this difficult equation? Suppose there exists a $y$ such that $$y \equiv \frac{d^n}{dx^n}e^{-x^2/2}$$. 
Prove that $$\frac{d^2y}{dx^2} + x\frac{dy}{dx} + (n+1)y = 0$$
I'm not sure where to start as Leibniz formula require at least 2 functions to begin with. There are no clear two functions in this problem.
My thought process:


*

*I could possibly factor out the y in all 3 terms, but this will make the derivatives invalid, wouldn't it?

*So since approach 1 wouldn't work, I could try integrating the whole equation, but I wouldn't get an exponential somewhere.

*I can try to sub in y into the equation, but I don't know how to proceed from here.


Using the product rule iteratively, we have 
\begin{eqnarray*}
x \frac{d^{n+1}}{dx^{n+1}} e^{-x^2/2} &=&\frac{d}{dx} \left(x \frac{d^n}{dx^n} e^{-x^2/2} \right) - \frac{d^{n}}{dx^{n}} e^{-x^2/2} \\
&=&\frac{d^2}{dx^2} \left(x \frac{d^{n-1}}{dx^{n-1}} e^{-x^2/2} \right) - 2\frac{d^{n}}{dx^{n}} e^{-x^2/2} \\
& \vdots & \\
&=&\frac{d^n}{dx^n} \left(x \frac{d}{dx} e^{-x^2/2} \right) - n\frac{d^{n}}{dx^{n}} e^{-x^2/2} .
\end{eqnarray*}
Now the equation can be rewritten as 
\begin{eqnarray*}
\frac{d^n}{dx^n} \left( \frac{d^2}{dx^2} e^{-x^2/2} +x \frac{d}{dx} e^{-x^2/2} -ne^{-x^2/2}+(n+1)e^{-x^2/2} \right)
\end{eqnarray*}
and it is easy to show that the content of the bracket is zero.
