Quote by Bipolarity
Also, why are we allowed to integrate both sides of the equation with respect to different variables?

You're not really integrating the y side in terms of y, because y and dy are functions of x. You're using substitution. For example, if [tex]y=C e^{k x}[/tex], then [tex]\frac{dy}{y}=\frac{C k e^{k x} dx}{C e^{k x}}=k dx[/tex].
As for the rest. I'm as lost as you are. I had a thread on this a long time ago. I'm looking for it now.