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What am I doing when I separate the variables of a differential equation?

My school textbook has a section on differential equations. One of the tricks used is the following-

$$\frac{dx}{dy}=\frac{x}{y}\implies\frac{dx}{x}=\frac{dy}{y} $$ Integration is then duly carried out.Sparation of the variables leaves an impression on me that somehow, $dy$ is "dividing" $dx$. Whereas,when I studied the definition of the derivative, it was like $$f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}.$$

I am not convinced how the so-called separation of variables is legal .Does it follow from the definition of the derivative ? Can anyone guide me to a proof?

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    Yeah I was bothered by this approach. One thing you could do is to treat $x,y$ as functions of one parameter $x(t),y(t)$ and then re-express everything in terms of $t$. – John Peter Jan 31 '13 at 20:04
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    When solving a differential equation, you can check your answer at the end. So even if you think the steps are invalid, go ahead! Actually, check your answer even if you think the steps are valid! – GEdgar Jan 31 '13 at 20:09
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    You have ${1\over x}{dx\over dy} ={1\over y}$, and you're assuming that $x$ is a function of $y$. From this you can conclude that an antiderivative of the left hand side of the above differs from an antiderivative of the righthand side by a constant. This is saying: $\int{1\over x}{dx\over dy},dy =\int{1\over y},dy$. But, by the chain rule, you can write the left hand side as $ \int {1\over x},dx$. The method of "separating variables" leads you, safely, to this conclusion. – David Mitra Jan 31 '13 at 20:21

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Here is a rough explanation: It follows because the derivative is a limit of a quotient of small values that, in fact, may be separated. That is, think of

$$\frac{dx}{dy}=\frac{x}{y}\implies\frac{\Delta x}{x} \approx \frac{\Delta y}{y}$$

Now imagine the act of integrating both sides in the latter equation as actually being one of summing both sides over all values of $x$ and $y$ in a given interval. This sum, as it turns out, will become a Riemann sum in the limit of extremely small steps $\Delta x$ and $\Delta y$, which of course leads to our integrals which we got from separation of variables.

Ron Gordon
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There's a notion called a "differential form". If $x$ and $y$ are functionally dependent and differentiable, then it turns out that the differential forms $dx$ and $dy$ are (more or less) multiples of each other, and the ratio happens to be $$ dx = \frac{dx}{dy} dy$$ This fact isn't really just cancelling the numerator and denominator.

Before I learned differential forms, I imagined a term "$dz$" as being a derivative with respect to some variable I hadn't decided yet. So I had interpreted the equation above as really meaning

$$ \frac{dx}{du} = \frac{dx}{dy} \frac{dy}{du} $$

where I hadn't really decided what variable I wanted to use for $u$.