Initially, Osgood used the integral ∫dr/f（r）for an unicity crite, rion to the differential equation y＇ = f（y）, f （0） = 0. The trivial solution is unique iff this integral goes to the infinite at the origin. Th...Initially, Osgood used the integral ∫dr/f（r）for an unicity crite, rion to the differential equation y＇ = f（y）, f （0） = 0. The trivial solution is unique iff this integral goes to the infinite at the origin. Then he can prove the unicity of the trivial solution of y＇ = y Ln｜Y｜, although the second member is not lipschitzian. Later, Bernfeld [1] shows that all the solutions of y＇ = f（y） do not explose iffthe same integral goes to the infinite at the infinite. Finally, we can adapt a result from the Cauchy works as follows： the trivial solution is a singular solution iffthe same integral vanishes at the origin. Using non standard analysis, we present the proofs of the different criterions and show that the Osgood integral was used by Cauchy before in the similar purpose.展开更多
文摘Initially, Osgood used the integral ∫dr/f（r）for an unicity crite, rion to the differential equation y＇ = f（y）, f （0） = 0. The trivial solution is unique iff this integral goes to the infinite at the origin. Then he can prove the unicity of the trivial solution of y＇ = y Ln｜Y｜, although the second member is not lipschitzian. Later, Bernfeld [1] shows that all the solutions of y＇ = f（y） do not explose iffthe same integral goes to the infinite at the infinite. Finally, we can adapt a result from the Cauchy works as follows： the trivial solution is a singular solution iffthe same integral vanishes at the origin. Using non standard analysis, we present the proofs of the different criterions and show that the Osgood integral was used by Cauchy before in the similar purpose.
