 Thus the dominant asymptotic behaviour of a solution to our ODE has been found,

By convention, the full asymptotic series is written as

so to get at least the first term of this series we have to take a further step to see if there is a power of out the front.Proceed by introducing a new subleading dependent variable,

and then seek asymptotic solutions for "C"("x").Substituting into the above ODE for "S"("x") we find
Repeating the same process as before, we keep and to find that

The leading asymptotic behaviour is then