For the differential equation, find (a) one solution and then (b) all solutions.
y" = -x-2
We need to think backwards twice. The first time, if we take the derivative of y' to get -x-2 we could have started with
Thinking backwards again, what function has as its derivative? This one:
y = ln x.
This is one solution. To find all solutions, start over. What has -x-2 as its derivative? Any function of the form
where B is any constant. In order to have such a y', we must have started with
y = ln x + Bx + C
where B,C are constants.