Van Dyke's matching rule I am looking at Perturbation Theory by E.J. Hinch. The author introduces Van Dyke's matching rule: (m term inner)(n term outer) = (n term outer)(m term inner) To be used to match the terms in an inner solution to the the terms in the outer solution for some perturbed differential equation. Hinch when comparing with introducing intermediate variables says: 'Van Dyke's matching rule usually works and is more convenient.' Does anyone know an example where this matching rule doesn't work? **Edit** In section $5.2.6$ of the book above, the author gives an example of where the matching rule is meant to fail. Is this an example where the matching rule actually fails or just shows some care in comparing terms is required? In particular, would the intermediate variables approach have any problems?

Okay I have found an example:

$f_{xx} + \frac{1}{x}f_x+f_x^2 +\epsilon ff_x=0$ for $x>1$

$f=0$ on $x=1$ and $f\rightarrow 1$

The solution is something close to $\log(1+\log(x)/\log(1/\epsilon))$

Van Dyke's rule fails because when expanding an infinite number terms are of the same order (in size).

However, with the intermediate variable approach since we are scaling by $\epsilon^{-\alpha}$ with $\alpha \in[0,1]$ we are able to choose $\alpha$ so that there are only a finite number of terms the same order.

The example is completely contrived though, I would feel much better with a more natural example.

Thank you for your effort J.Meyer and J.M.