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Sorry for asking such stupid things. But I have this simple question. In some books, instead of solving $dy / dx$ they just solve $dx / dy$ by raising all to the power $-1$. For example in this exercise in a directory they do this: instead of solving $$\frac{dy}{dx}= \frac {1}{x\sin y + 2\sin 2y}$$ which is not linear, they solve this: $$\frac{dx}{dy}= x \sin y + 2 \sin 2y$$ which is linear in $x$. The question is, can we always do this? How is it justified?

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I have tried to keep much of the original language. The formulas are my interpretation of what was intended. Please look at how the LaTeX was done. Also, please tell me if the two main displayed formulas are not the ones you want. –  André Nicolas Aug 27 '11 at 22:48
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up vote 3 down vote accepted

If the question is, is it true that $dx/dy=1/(dy/dx)$, then, yes, it is true. See, e.g., this Wikipedia link.

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