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By Bolton and Paul Ltd, Manufacturer

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7) which indeed provides one first integral [390] K1 = (x2 − 2σ z)e2σ t , b = 2σ . 15)), they all have an even parity1 and it can be proven [283] that they are the only ones in the polynomial class. 19), and an interesting open problem is to find a first integral, if any, associated to the second solution b = 1 − 3σ of Q2 = 0. In order to find first integrals of dynamical systems, an original approach has been developed [40], which has no relation with singularities but which takes advantage of basic properties in linear algebra.

105) (n) (0) (n−1) + R (x, u , . . 40), and R(1) is identically zero. 106) without movable critical points. 2). 1). The two main implementations of this perturbation are the Fuchsian perturbative method [149, 82], described hereafter, and the nonFuchsian perturbative method [318], described in the next section. The Fuchsian perturbative method is adapted to the presence of negative integer indices in addition to the ever present value −1, and it generates additional no-log conditions. 108) represents a particular solution containing a number of arbitrary coefficients equal to the number of Fuchs indices, counting their multiplicity.

0 -8 -6 -4 -2 0 2 4 6 8 Fig. 5. ⎧ M M2 c2 ω c 2 q ⎪ ⎪ ⎨ − − ϕ − + M + + 2 = 0, 2 2M 4M 2p p p 4p c M ⎪ ⎪ ⎩ϕ + ϕ − = 0. 49) According to the results in Sect. 58). The indices −1 and 0 correspond to the two “angles” ξ0 , ϕ0 (the irrelevant origins of ξ and ϕ ), and the two others (3, 4) to two “actions” to be found. 49) admits the integrating factor M, resulting in the first integral Mϕ − c M = K1 . 50) Since this relation is homographic, it is equivalent to consider either ϕ or M. 49) admits the integrating factor M , resulting in another first integral c2 − 4 ω p M 2 2q 2 K12 + M + = K2 .

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1881 Catalogue of Glasshouses by Bolton and Paul Ltd, Manufacturer

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