Asymptotics have been much used in the study of differential equations. The method of undetermined coefficients is one common technique. At its most basic, this consists of substituting a general power series into the equation and then comparing terms in order to find the coefficients. The same idea can be made to work with series in other base functions, and also series where several different base functions may appear, for example x and log x A drawback is that one has to know in advance which base functions are going to be needed. There is a great fund of expertise on such questions, especially for linear ordinary differential equations; see [119] for example. In this chapter, we look at what can be done for non-linear equations using nested expansions.