I'm supposed to find the solution of
The solution in the book manipulates it to
and proceeds to use an integrating factor to solve it to
Which makes sense, but I did it by isolating the differential of y, substituting , and simplifying it to
which gives, on solving,
Which isn't the same.
I can't see any flaw in what I did except for dividing variables throughout the equation (which I see everywhere in methods of solving differential equations, without a clue as to why.)
Can anyone help me with where I got it wrong?
EDIT: I'll add some detail on my steps;