Why can we "separate" variables? > **Possible Duplicate:** > 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?

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 $u$.