Notation for separation of variables

[lLovePhysics]

My book has a problem that requires you to separate variables (one side has all the y terms and one side has all of the x terms):

$$\sin{xy'}=\cosx$$

Equation after separation of variables:

$$dy=\cot{x}dx$$

My question is, how do you know that the y' is contained within the sine function or out of it? Are the derivatives always written after and outside of the preceding function?

 

[quasar987]

There is no way to know for certain what is in the sin and what is not, and I am often wondering myself. But often, it is clear from context. In your case for instance, if the y' were in the sin, the equation would not be separable.

 

[neutrino]

$$\sin{\left(xy'\right)}$$

and

$$\sin{\left(x\right)}y'$$

The parentheses makes all the difference. It really shouldn't matter on which side the derivative is written. It would be a bad book, if no such unambiguous distinction is made, in my opinion. But looking at the example you provided, it appears that they went you to assume that it's a product of function of x and the derivative of y, unless it's stated as $\sin{\left(xy'\right)}$.