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On completely integrable implicit ordinary differential equations
| Title: | On completely integrable implicit ordinary differential equations |
| Authors: | Takahashi, Masatomo Browse this author |
| Keywords: | implicit ordinary differential equation | | geometric solution | | singular solution | | complete solution | | Clairaut type | | reduced type |
| Issue Date: | 23-Jul-2010 |
| Publisher: | Department of Mathematics, Hokkaido University |
| Journal Title: | Hokkaido University Preprint Series in Mathematics |
| Volume: | 964 |
| Start Page: | 1 |
| End Page: | 20 |
| Abstract: | For smooth explicit n-th order ordinary differential equations, there exists a unique solution with initial condition and hence exists an n-parameter family of solutions at least locally. On the other hand, for smooth implicit ordinary differential equations, existence and uniqueness for solutions with initial condition does not hold. In this paper, we give a necessary and sufficient condition for existence of an n-parameter family of geometric solutions in the smooth category. Moreover, we give a sufficient condition that implicit ordinary differential equations have a unique geometric solution with initial condition. As a consequence, we classify completely integrable first and second ordinary differential equations in detail. |
| Type: | bulletin (article) |
| URI: | http://hdl.handle.net/2115/69771 |
| Appears in Collections: | 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics
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Submitter: 数学紀要登録作業用
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