Solved: The Tautochrone. A problem of interest in the history of mathematics is that of

Chapter 6, Problem 29

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The Tautochrone. A problem of interest in the history of mathematics is that of finding the tautochrone4the curve down which a particle will slide freely under gravity alone, reaching the bottom in the same time regardless of its starting point on the curve. This problem arose in the construction of a clock pendulum whose period is independent of the amplitude of its motion. The tautochrone was found by Christian Huygens (1629 1695) in 1673 by geometrical methods, and later by Leibniz and Jakob Bernoulli using analytical arguments. Bernoullis solution (in 1690) was one of the first occasions on which a differential equation was explicitly solved. The geometric configuration is shown in Figure 6.6.2. The starting point P(a, b) is joined to the terminal point (0, 0) by the arc C. Arc length sis measured from the origin, and f(y) denotes the rate of change of s with respect to y:f(y) = dsdy =1 +dxdy21/2. (

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