[Community Question] Calculus: What to multiply by to get correct form ODE

One of our user asked:

Suppose $y'' + f(x)y = 0$ where $M \geq f(x) \geq m > 0$ on some interval $[a,b]$, then the number zeros $N$ of a non trivial solution is $\lfloor\frac{(b-a)\sqrt{m}}{\pi}\rfloor \leq N \leq \lceil\frac{(b-a)\sqrt{M}}{\pi}\rceil$

Simple.

Now suppose I have an equation $y''+4y'+\frac{8x+\sin(x)}{x+1}y = 0$ and I want to estimate the number of zeros of a non trivial solution.

I can't use the theorem as is, because the ODE is not in the correct form, to fix this, we can multiply by $e^{2x}$ and get $y''e^{2x}+4y'e^{2x}+\frac{8x+\sin(x)}{x+1}ye^{2x} = 0$

Now if we let $ye^{2x} = z$ we have an ODE $z'' + (\frac{8x+\sin(x)}{x+1}-4)z = 0$ which is in the correct form

How did the professor know to multiply by $e^{2x}$? Is there a method to this or was this just a lucky guess

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Order of elements of the Prüfer groups $\mathbb{Z}(p^{\infty})$

Let $\mathbb{Z}(p^{\infty})$ be defined by $\mathbb{Z}(p^{\infty}) = \{ \overline{a/b} \in \mathbb{Q}/ \mathbb{Z} / a,b \in \mathbb{Z}, b=p^i$ $ with$ $ i \in \mathbb{N} \}$ , I wish show that any element in $\mathbb{Z}(p^{\infty})$ has order $p^n$ with $n \in \mathbb{N}$ . i try several ways but I have not been successful, some help ?? thank you

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