Qualitative analysis of $y' = \frac {(y+1) \cdot \sinh(x)}{(y+1) \cdot \log(y+1) +y}$

Question

Good evening,

I'd like to ask a very specific question about a method that my professor used within the following exercise :

• Qualitative analysis of $ y' = \frac {(y+1) \cdot \sinh(x)}{(y+1) \cdot \log(y+1) +y}$

• Domain : $ y > -1$, $y \ne 0$

After Integration by separable variable i reach the following result :

• $y\cdot \log(y+1) = \cosh(x) + C$.

From now on I'd like to know the name of the method, or at least understanding the concept of what my professor did, because it seems that he created a sort of two distinct graphics (one for $x$, one for $y$) that have a sort of one-to-one correspondence which gathered, help me build my real graphic.

I'll attach the exercise as clear as possible.

Thanks.