Ordinary Differential Equations
The first two chapters of this book have been thoroughly revised and sig nificantly expanded. Sections have been added on elementary methods of in tegration (on homogeneous and inhomogeneous first-order linear equations and on homogeneous and quasi-homogeneous equations), on first-order linear and quasi-linear partial differential equations, on equations not solved for the derivative, and on Sturm's theorems on the zeros of second-order linear equa tions. Thus the new edition contains all the questions of the current syllabus in the theory of ordinary differential equations. In discussing special devices for integration the author has tried through out to lay bare the geometric essence of the methods being studied and to show how these methods work in applications, especially in mechanics. Thus to solve an inhomogeneous linear equation we introduce the delta-function and calculate the retarded Green's function; quasi-homogeneous equations lead to the theory of similarity and the law of universal gravitation, while the theorem on differentiability of the solution with respect to the initial conditions leads to the study of the relative motion of celestial bodies in neighboring orbits. The author has permitted himself to include some historical digressions in this preface. Differential equations were invented by Newton (1642-1727).
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amplitude approximation axis basis called characteristic equation chart circle coefficients complex Consider constant Corollary corresponding cycle defined Definition denote depends derivative diffeomorphism differential equation dilations direction field domain eigenvalues eigenvector energy equation of variations equilibrium position equivalent Euclidean space Euler example exists exponential extended phase space Find finite first-order formula function graph homogeneous inhomogeneous equation initial condition instant integral curves interval ip(t Lemma linear equation linear operator linear transformation manifold mapping matrix motion neighborhood Newton nonzero obtain one-parameter group parameter phase curves phase flow phase plane phase point phase space phase velocity phase velocity vector polynomial Problem Proof Prove quasi-homogeneous quasi-polynomials rectifying Remark right-hand side rotation Sect sequence singular points smooth solution ip solution of Eq solve space Rn sphere stable subspaces sufficiently small tangent topological variables vector field vector space Wronskian zero