Thread: Reduction of order, dependent variable missing trouble

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.

The two answers are, in fact, the same. Starting from your we can "multiply and divide" by C to get

Makes sense now that I see it done, thanks. I would have never thought to do the +1 -1 thing.