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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.85 Special Functions

T. W. Körner

Suppose that the only functions we have come across are quotients of polynomials and that we are asked to solve the differential equation

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for all x > 0, subject to the condition f(1) = 0.

If we try f(x) = P(x)/Q(x), where P and Q are polynomials with no common factors, then we find that

Image

By comparing coefficients we can show that Q(0) = P(0) = 0, which shows that, contrary to our assumptions, both P(x) and Q(x) are divisible by x. Thus, we cannot solve equation (1) in terms of known functions. However, THE ...

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