On Implicit Second-Order Ordinary Differential Equations: Completely Integrable and Clairaut Type

Takahashi, Masatomo

doi:10.1007/s10883-007-9013-9

Cited by 4 publications

(8 citation statements)

References 11 publications

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“…It follows that F = 0 is completely integrable at z 0 by Theorem 1.1(1). We conclude by Proposition 3.2 in [11] that F = 0 is of Clairaut type at z 0 .…”

Section: Example 12 (Second Order Classical Clairaut Equations Withsupporting

confidence: 51%

“…Proposition 2.2). The Clairaut type has already appeared in [11] as a necessary and sufficient condition for existence of a smooth complete solution. Moreover, we consider properties of completely integrable equations.…”

Section: Example 12 (Second Order Classical Clairaut Equations Withmentioning

confidence: 99%

“…In this paper, the following notions are basic [1,2,[4][5][6]11]. Following the definition of parametrized version for smoothness of classical solutions, a smooth complete solution of F = 0 at z 0 is defined to be a two-parameter family of smooth function germs y = f (t, r, s) such that…”

Section: Introduction An Implicit Second Order Ordinary Differentialmentioning

confidence: 99%

“…Here we give an example illustrating the notions of a complete solution and a complete singular solution. In [11], we discussed second order ordinary classical Clairaut equations given by…”

Section: Introduction An Implicit Second Order Ordinary Differentialmentioning

confidence: 99%

“…This is an example of a (smooth) completely integrable second order ordinary differential equation (see [8][9][10][11][12] and Example 4.1 for details). We now consider the following case.…”

Section: Introduction An Implicit Second Order Ordinary Differentialmentioning

confidence: 99%

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On complete solutions and complete singular solutions of second order ordinary differential equations

Takahashi

2007

Colloq. Math.

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Abstract. A complete solution of an implicit second order ordinary differential equation is defined by an immersive two-parameter family of geometric solutions on the equation hypersurface. We show that a completely integrable equation is either of Clairaut type or of first order type. Moreover, we define a complete singular solution, an immersive one-parameter family of singular solutions on the contact singular set. We give conditions for existence of a complete solution and a complete singular solution of implicit second order ordinary differential equations.

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“…It follows that F = 0 is completely integrable at z 0 by Theorem 1.1(1). We conclude by Proposition 3.2 in [11] that F = 0 is of Clairaut type at z 0 .…”

Section: Example 12 (Second Order Classical Clairaut Equations Withsupporting

confidence: 51%

Section: Example 12 (Second Order Classical Clairaut Equations Withmentioning

confidence: 99%

Section: Introduction An Implicit Second Order Ordinary Differentialmentioning

confidence: 99%

“…Here we give an example illustrating the notions of a complete solution and a complete singular solution. In [11], we discussed second order ordinary classical Clairaut equations given by…”

Section: Introduction An Implicit Second Order Ordinary Differentialmentioning

confidence: 99%

Section: Introduction An Implicit Second Order Ordinary Differentialmentioning

confidence: 99%

See 3 more Smart Citations