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

Conversely, if y xx (x) = F ( x, y(x), y x (x) ) is equivalent, through a diffeomorphism(x, y) ↦→ (x ′ , y ′ ), to yx ′ ′ x ′(x′ ) = 0, then its submanifoldof solutions y = Q(x, a, b) is transformed to y ′ = Q ′ (x ′ , a, b) and sincey x ′ ′ x ′(x′ ) = 0, the function Q ′ is necessarily of the form:y ′ = b ′ (a, b) + x ′ a ′ (a, b).Because the condition of solvability with respect to the parameters is invariant,the rank of (a, b) ↦→ ( a ′ (a, b), b ′ (a, b) ) is again equal to 2, whichconcludes the proof.Coming now back to the wanted characterization of sphericality, our moregeneral goal now amounts to characterize, directly in terms of its fundamentalsolution function Q(x, a, b), the local equivalence to y ′ x ′ x ′(x′ ) = 0 of asecond-order ordinary differential equation y xx (x) = F ( x, y(x), y x (x) ) .Afterwards at the end, it will suffice to replace Q(x, a, b) simply byΘ(z, z, w) in the obtained equations.But before going further, let us explain how a certain generalized projectiveduality will simplify our task, as already said in the Introduction. Thus,let (E ): y xx (x) = F ( x, y(x), y x (x) ) be a differential equation as abovehaving general solution y = Q(x, a, b) = −b+xa+O(x 2 ), with initial conditionsb = −y(0) and a = y x (0). The implicit function theorem enables usto solve b in the equation y = Q(x, a, b) of the associated submanifold ofsolutions M (E) in terms of the other quantities, which yields an equation ofthe shape:b = Q ∗ (a, x, y) = −y + ax + O(x 2 ),for some new local K-analytic function Q ∗ = Q ∗ (a, x, y). Then similarly aspreviously, we may eliminate x and y from the two equations:b(a) = Q ∗ (a, x, y) = −y + ax + O(x 2 )b a (a) = Q ∗ a (a, x, y) = x + O(x2 ),that is to say: x = X ( a, b(a), b a (a) ) and y = Y ( a, b(a), b a (a) ) , and we theninsert these two solutions in:b aa (a) = Q ∗ aa (a, x, y)(= Q ∗ aa a, X(a, b(a), ba (a)), Y (a, b(a), b a (a)) )=: F ∗( a, b(a), b a (a) ) .We shall call the so obtained second-order ordinary differential equation thedual of y xx (x) = F ( x, y(x), y x (x) ) .In the case of a hypersurface M ⊂ C 2 , solving w in the equation w =Θ(z, z, w) gives nothing else but the conjugate equation w = Θ(z, z, w),203