How to eliminate the first order term from a second order ODE

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

and

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 / a x.