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The Equation f(2x) = 2f(x)f′(x)

A. F. Beardon
Mathematics Magazine
Vol. 86, No. 1 (February 2013), pp. 48-52
DOI: 10.4169/math.mag.86.1.048
Stable URL: http://www.jstor.org/stable/10.4169/math.mag.86.1.048
Page Count: 5
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The Equation f(2x) = 2f(x)f′(x)
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Abstract

Summary We discuss solutions of the equation f(2x) = 2f(x)f′(x), which is essentially a delay differential equation, with the boundary condition f–(0) = 1, on the interval [0, + ∞). In particular, we note that the only known solutions of this type that are positive when x is positive are the functions c−1 sinh(cx), where c > 0, and the function x.

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