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and so satisfies

with The general solution of this equation can be found, albeit in implicit
form, as

where and A is an arbitrary constant which is
evaluated as

when the boundary condition on X = 0 is imposed. As (which will give the
behaviour outside the boundary layer), we see that

which therefore automatically matches with the solution already found in the region
away from the boundary layer, (5.151). It is left as a straightforward exercise to find
higher-order terms, or to write down a composite expansion valid for see
§1.10.

This first example has presented us with a very routine exercise in elementary
boundary-layer theory; the next has a similar structure, but in only one of the two
components.

E5.27 Enzyme kinetics
A standard process in enzyme kinetics concerns the conversion of a substrate (x) into
a product, by the action of an enzyme, via a substrate-enzyme complex (y); this is the
Michaelis-Menton reaction. A model for this process is the pair of equations

where and are positive constants and the initial conditions are

The parameter measures the rate of the production of y; we consider the problem
posed above, with It is clear that we should expect a boundary-layer structure