Authors will often say that “without loss of generality” they will assume that a differential equation has no first order derivative term. They’ll explain that there’s no need to consider
because a change of variables can turn the above equation into one of the form
While this is true, the change of variables is seldom spelled out. I’ll give the change of variables explicitly here in case this is helpful to someone in the future. Define u(x) and r(x) by
With this change of variables
Proof: Calculate u” + ru and use the fact that y satisfies the original differential equation. The calculation is tedious but routine.
Example: Suppose we start with
Then dividing by x we get
Now applying the change of variables above gives
and our original y is u / √ x.
Update: Here is a new post generalizing the content of this post to higher-order ODEs.