# Is this differential equation separable??

Just a quick question... I have the equation:

$$\frac{dw}{dy} w^{-2}=0$$

Is this separable? i.e. can I go :

\begin{align} \frac{dw}{dy} w^{-2}&=0 \\[8pt] (w^{-2}) \,dw&=(0)\,dy \tag{*} \\[8pt] -w^{-1}&=A \end{align} and thus $$w = B$$ with $A,B$ being arbitrary constants with $B \not= 0$

Is this ok? The only thing I'm wondering about is the starred line... can I separate the equation like that with only a zero on the RHS?

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Why not multiply by $w^{-2}$ and get $dw/dy = 0$ ... – GEdgar Sep 5 '12 at 20:35
@GEdgar Wasnt sure if I wouldn't lose generality by doing that though I did think about it. I see it would give the same result though. So either idea is valid then ? – Ronald Sep 5 '12 at 20:39
....or rather by $w^2$. – Michael Hardy Sep 5 '12 at 23:50

The reason why you're not used to applying separation of variables to a situation like that is because it's actually unnecessary: you have the product of two terms equalling zero, so one of them must be zero. $w^{-2}$ is never zero, regardless of $w$, so $\frac{dw}{dy}$ is zero. So $w$ is constant.