![]() The Sturm Comparison Theorem5 Power Series Solutions and Special Function Oscillations and the Sturm Separation Theore Newton4 Qualitative Properties of Solution Operator Methods for Finding Particular SolutionsAppendix A. Newton's Law of Gravitation and the Motions of the Planet Vibrations in Mechanical and Electrical System The Homogeneous Equation with Constant Coefficient The Use of a Known Solution to Find Anothe The General Solution of the Homogeneous Equatio Simple Electric Circuits3 Second Order Linear Equation Fermat and the Bernoullis2 First Order Equation Growth, Decay, Chemical Reactions, and Mixin Preface to the Second EditionPreface to the First EditionSuggestions for the Instructor1 The Nature of Differential Equations.All in all, it is a beautiful family of curves based on beautiful mathematics and a rich and fascinating history. More recently, the major focus has been on ecient numerical techniques for computing the elastica and fitting it to spline problems. It is also the mathematical model of the mechanical spline, used for shipbuilding and similar applications, and directly insipired the modern theory of mathematical splines. In addition, it has a number of analogies with physical systems, including a sheet holding a volume of water, the surface of a capillary, and the motion of a simple pendulum. The elastica can be understood from a number of dierent aspects, including as a mechanical equilibrium, a problem of the calculus of variations, and the solution to elliptic integrals. The complete solution is most commonly attributed to Euler in 1744 because of his compelling mathematical treatment and illustrations, but in fact James Bernoulli had arrived at the correct equation a half-century earlier. This report traces the history of the elastica from its first precise formulation by James Bernoulli in 1691 through the present. ![]()
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