We have started PDEs in our MAth course in BE ME. We are following Kreyszig and I had problem understanding the basic concepts. It says "PDEs SOLVABLE AS ODEs : It happens if a PDE involves derivatives with respect to one variable only (or can be transformed to such a form), so that the other variable9s) can be treasted as parameters(s)"

There's an example I am confused in,

u_xy = -u_x where

The steps goes as,

,

,

,

** and by integration with respect to x, ** ** where **
I do not understand the bold part and the natural log step how does

come in ?