Etude algebrique et algorithmique des singularites des by Hubert E.

By Hubert E.

Show description

Read or Download Etude algebrique et algorithmique des singularites des equations differentielles implicites PDF

Similar mathematics books

Additional resources for Etude algebrique et algorithmique des singularites des equations differentielles implicites

Sample text

7 Proposition: Let R be a di erential ring. Whenever R contains a eld isomorphic to Q , the radical of a di erential ideal of R is a di erential ideal, and thus a radical di erential ideal. 2 . For an element a of R and an integer we show by induction that for any r, 0  r  , 1, Proof: Consider any derivation 2 N qr =  , 1 : : :  , rp ,r,1  This is true for r = 0 since q0 = p ,1 p = p Assume qr 2 p for 0  r , 1. Then qr = p2r+1 2 p 2p :  , 1 : : :  , r , 1p ,r,2 p2r+1 +2r + 1  , 1 : : :  , rp ,r,1 p2r 2p 2 p and therefore q qr = qr+1 +2r +1qr 2p 2 p ; which drives us to the desired conclusion.

T. every other element of A. Such a set is nite and triangular : no pair of di erential polynomials in A have the same leader. t. t. any element of A. Let A be an auto-reduced set. t. 9 . t. A satisfying 1. When A is an auto-reduced set of FfY g, we will note mathbfhA the product of all initials and separants of the di erential polynomial in A. Characteristic sets A ranking on FfY g induces a pre-order on the set of all auto-reduced subsets of FfY g. Let A = a1 ; : : : ; ar and B = b1 ; : : : ; bs be two auto-reduced subsets.

40 d . 1 y~1 x et y~2 x, pour c = 0 et a; b = 0; 0;  13 ; 13 ;  31 ; , 241  Exemple : Soit l'
equation di erentielle 6 y y00 , 5 y023 + 729 y4 = 0: dont l'unique solution singuli ere est y0x = 0. La solution g
en
erale peut ^etre donn
ee par 3 y2x = x , a + b x , a2 o u a et b sont les constantes arbitraires. y0x = 0 est donc une enveloppe des courbes de la solution g
en
erale et le contact est d'ordre 3. Nous repr
esentons quelques unes de ces solutions. 1. 02 a = ,1; 0; 1, b = 1 2 42 d .

Download PDF sample

Rated 4.76 of 5 – based on 8 votes