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I have been learning about differential equation on Khan Academy.

https://www.youtube.com/watch?v=DL-ozRGDlkY

And at 1:57 Sal multiplied the whole equation by dx and cancelled the dx in dy/dx. But isn't dy/dx something like an operator. Is it mathematically correct to do such cancellation?

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    See http://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio – 2017-02-26

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So our good 'ol Sal is dealing with separable differential equations. The differential operator $\frac{dx}{dt}$ is indeed the same as $\frac{dx}{1} \cdot \frac{1}{dt}$, which means you could indeed cancel it out by multiplying both sides by one or the other. This is done because then each side may be integrated to reach a solution. As is $dx$ or $dt$ in the integral, $D=\frac{dx}{dt}$ is simply a value, and can be manipulated like any other variable.

I hope this helps!

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    I'm not sure I agree with this. How is the operator split into a product? – 2017-02-26
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    A ratio of differentials (can) make sense; a ratio of a differential to a scalar does not make sense. – 2017-02-26