Determining Linearity in First-Order Differential Equations

[NJJ289]

So I just started taking an intro diff eq course and here's one of my homework problems:

"Determine whether the given first-order diff eq is linear in the indicated dependent variable."

(y²-1)dx + xdy=0; in y; in x

I got the whole bit about the general form for linearity but I was thrown off by having just a 'dx' and 'dy' instead of the more familiar 'dy/dx'

(answer to the question is non linear when y is dependant, linear when x is dependant)

I'm confused as to what exactly 'dy' or 'dx' means, both conceptually and mathematically. I have a feeling there's a nice thread on this somewhere...

thanks for the help!

 

[Studiot]

Please note forum policy on homework.

https://www.physicsforums.com/showthread.php?t=88061

Having said that here is a hint.

$$\begin{array}{l} \left( {{y^2} - 1} \right)dx = - xdy \\ \left( {1 - {y^2}} \right)dx = xdy \\ \frac{{dy}}{{dx}} = \frac{{\left( {1 - {y^2}} \right)}}{x} \\ \frac{{dx}}{{dy}} = \frac{x}{{\left( {1 - {y^2}} \right)}} \\ \end{array}$$

Can you see which has x as the dependent variable and which has y?