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5) The following problem was first posed in 1638, and worked on, amongst others by Descartes, Fermat, Leibniz and Jakob Bernoulli until 1693; so you should have no trouble solving it in a week! Its a geometry problem; find the formula of a curve y(x) such that the length of the segment of the tangent to the curve between the curve and the r-axis is equal to a constant a at all points on the curve. We draw a right-angled triangle with base on the z-axis, and height y and hypotenuse a: from which we see that the differential equation of the curve must be dy dr a2 where a > 0 is an arbitrary constant (the length of the tangent segment). Find the solution in implicit form. Can you write the solution as a g(w), that is z equal to a function of y? Hint: The substitution 2 - v may be uscful to evaluate one of the integrals that arises
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