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As expansion (2.44) breaks down where or
and then we introduce the scaled variables

and then the equation in (2.41) becomes

For this equation, it is clear that we must seek a solution in the form

and then (2.45) yields

and so on. The first equation here can be written as

where is an arbitrary constant; thus

and both and the choice of sign are to be determined by matching.
From the first term in (2.44) we obtain

and from (2.47) we have

which matches with (2.48) only for the positive sign and then with Thus
(2.47) becomes