On Certain Transformations Not Changing the Integral Curves of a Differential Equation of the First Order |
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abscissa arbitrary B₁[y bounded Consequently contact-free curves continuous functions continuous periodic function continuous periodic solution Corollary curve of equation curve y Y1(x differentiable function differential equation dx dy e-kx equa equation 24 exact differential expression Y dx find a number fixed sign following theorem func function X depends function y(x Green's formula inequality 9 integral curve interval leaves the region logarithmic derivatives meaning monotone nonoscillatory solutions obtain operator A[y operator B[y P₁ passing periodic continuous solutions periodic function point a,yo(a point in common positive constant proof region G Riccati equation satisfies the inequality satisfying the condition sequence yn(x side of equation solution continuous solution of equation solution y1(x solutions of 24 tions y(x transforms the curve uniform convergence valid write the operator x-axis X₁ Y₁ x y₁(x Y₂ Yk(x yo(a όχ ΟΧ ΟΥ ду дх