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abscissa belong Consequently contact-free curves continuous functions continuous periodic function continuous periodic solution convex set Corollary curve of equation defined differentiable function differential equation equa equal equation 24 equation F(x,y exact differential find a number fixed sign following theorem form A[y func function with period function X depends function X(x,y functions y^(x Green's formula holds inequality 9 inequality dx integral curve integral vanishes interval leaves the region leaving the point logarithmic derivatives mean value theorem nonoscillatory solutions obtain operator A[y operator B[y operator has meaning periodic continuous solutions point a,yQ(a point in common proof region G satisfies the inequality satisfying the condition sequence iyn(x sequence y(a side of equation solu solution continuous solution of equation solution y^(x solutions of 24 suppose Theorem XII tions y^(x transforms the curve uniform convergence valid write the operator x-axis Ydx-Xdy yi(x ym(x yn(x yo(x yQ(x