Are Differential equations without "$x$" considered separable?

Question

For example: Do we consider all differential equations of the form $y'(x)=f(y)$ separable? why/why not?

Answer 1

It would be separable. Check if $f(y) = 0$ is a solution. Then ignore this case to get $$\frac{y'}{f(y)} = 1 $$ so your solution looks like $$\int\frac{y'}{f(y)} = x + C $$

Answer 2

I am interpreting what you have written as:

$$y'(x)=f(y(x))$$

Separable equations can be written in the form:

$$A(y(x))y'(x) = B(x)$$

So if you choose $A(y) = \frac{1}{f(y)}$ and $B(x) = 1$. $f(y)$ should not equal zero on your interval of analysis in order for you to be able to make this transformation.