Hi I've been having trouble with a reduction of order problem from my textbook. By making the appropriate substitutions of for the missing dependent variable , separating variables and integrating I can get the equation in the below form

My main problem from here is I can't seem to solve for u in such a way as to make the next integration come out to the answer given in the book. After struggling with this for a while, I actually found the exact same problem in a Differential Equations Demystified book I purchased. They go in one step from what is above to

How did they do this? And is there any intuition behind why you would decide to solve for u in this way as opposed to what I tried to do which was more along the lines of

Thanks for any help.