Problem #2 - the Y 2n functions

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Abstract
Many branches of mathematical physics use equations of the form[EQUATION]where λ is a small quantity, and the primes denote differentiation with respect to x. In the absence of a general solution, one tries to write f(x) as an expansion in powers of λ. More neatly, if[EQUATION]is tried, to fit the structure of (1), then the work reduces to the derivation of a series expansion for q(x). The solution is [1][EQUATION]where N signifies the order of approximation to which one wishes to go, and Y 2n represents the member of order λ 2n of a family of functions obtained by substitution of (3) into (1) and (2). The problem is to compute Y 2n for as many values of n as possible.

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