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Equations By F R Moulton and W D McMillan
Solutions of Differential Equations as Power Series in Parameters
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analytic continuation arbitrary constants becomes Celestial Mechanics characteristic equation characteristic exponents closed orbits coefficients computation Consequently consider constants of integration contain coordinates corresponding cosine series crosses the x-axis defined differential equations distinct from zero equal to zero existence expansible as power expressions factor finite bodies finite masses follows four equations fundamental set Hence highest multiple identically zero initial conditions initial values integer Jacobian last equation linear terms motion obtained odd function odd multiples orbits of Class orbits of ejection parameter period 2ir period 2t periodic functions periodic orbits periodic solutions periodicity condition plane positive power series properties purely imaginary right members second equation set of solutions shown sines of odd solution of equations solved spherical pendulum substituting sufficiently small sum of cosines sum of sines Suppose terms independent tions transformation two-body problem uniquely determined unity vanish x-axis perpendicularly