Movable Singularities of a Class of Nonlinear Ordinary Differential Equations of Arbitrary Order

In this paper we consider nonlinear ordinary differential equations   y ^{( n )}= F ( y ', y , x )  of arbitrary order   n ≥ 3  , where F is algebraic in   y , y '  and locally analytic in x . We prove that for   n > 3  these equations always admit movable branch points. In the case   n = 3  these equations admit movable branch points unless they are of the known class   y '= a ( x )( y ')²+ ( b ₂( x ) y ²+ b ₁( x ) y + b ₀( x )) y '+ ( c ₄( x ) y ⁴+ c ₃( x ) y ³+ c ₂( x ) y ²+ c ₁( x ) y + c ₀( x ))  , where   a ,  b_j ,  c_j   are locally analytic in x .