Introduce the change of coordinates
So is now a function of . Recompute the derivatives with respect to these new coordinates.
Plug these derivatives into the original equation.
Choose any values of to make the problem easy to solve, with the caveat that . For example, let so that the term goes away. Then let so that the coefficient equals unity. Then,
Use the integrating factor
Here is an undetermined function of .
Now we determine the unique solution that corresponds to the inital condition . Using the general solution above, set .
But in the general solution, we have instead of . So making the appropriate substitutions,
Plug this function into the general solution.