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which, from (5.105), must therefore satisfy the equations

and then (5.107b) gives

where is an arbitrary function. We may impose the initial conditions, (5.106a,b),
and so then (5.107a) becomes simply

and the (similarity) solution which satisfies (5.106a,c,e) is

(provided that t = 0 is interpreted as Thus

which does not satisfy the boundary value on x = 0 and so we require
the boundary layer near here.
Let us introduce and write

then equations (5.105) become

the leading-order problem (zero subscript) therefore satisfies

The solution of this pair is to satisfy the matching conditions