Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

200to yield both a representation for a and and a representation for b of the form:[ (a = A x, y(x), yx (x) )(7.28)b = B ( x, y(x), y x (x) ) ,for certain two local K-analytic functions A and B of three independentvariables (x, y, y x ), that one may insert afterwards in the second order derivative:y xx (x) = Q xx(x, a, b)= Q xx(x, A(x, y(x), yx (x)), B(x, y(x), y x (x)) )=: F ( x, y(x), y x (x) ) ,which yields the differential equation (E M ) associated to the submanifoldM solvable with respect to the parameters. In summary:Proposition. ([19]) There is a one-to-one correspondence:(E M ) = (E ) ←→ M = M (E )between second-order ordinary differential equations (E ) of the generalform:y xx (x) = F ( x, y(x), y x (x) )and submanifolds (of solutions) M of equation:y = Q(x, a, b)that are solvable with respect to the parameters, and this correspondencesatisfies:(EM(E))= (E ) and M(EM ) = M.We now claim that solvability with respect to the parameters is an invariantcondition, independently of the choice of coordinates. Indeed, lety = Q(x, a, b) be any submanifold of solutions, call it M , and let:(x, y, a, b)↦−→(x ′ (x, y), y ′ (x, y), a, b )be an arbitrary local K-analytic diffeomorphism fixing the origin( which leaves untouched the parameters. The vector of coordinates1, Qx (x, a, b), 0, 0 ) based at the point ( x, Q(x, a, b), a, b ) of M is sent,through such a diffeomorphism, to a vector whose x ′ -coordinate equals:ddx[x ′ (x, Q) ] = x ′ x +Q x x ′ y . Therefore the implicit function theorem insuresthat, provided the expression:x ′ x(x, y) + Q x (x, a, b) x ′ y(x, y) ≠ 0does not vanish, the image M ′ of M through such a diffeomorphism canstill be represented, locally in a neighborhood of the origin, as a graph of asimilar form:y ′ = Q ′( x ′ , a, b ) ,