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- Singular_solution abstract "A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full real line. Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions.In some cases, the term singular solution is used to mean a solution at which there is a failure of uniqueness to the initial value problem at every point on the curve. A singular solution in this stronger sense is often given as tangent to every solution from a family of solutions. By tangent we mean that there is a point x where ys(x) = yc(x) and y's(x) = y'c(x) where yc is a solution in a family of solutions parameterized by c. This means that the singular solution is the envelope of the family of solutions.Usually, singular solutions appear in differential equations when there is a need to divide in a term that might be equal to zero. Therefore, when one is solving a differential equation and using division one must check what happens if the term is equal to zero, and whether it leads to a singular solution. The Picard–Lindelöf theorem, which gives sufficient conditions for unique solutions to exist, can be used to rule out the existence of singular solutions. Other theorems, such as the Peano existence theorem, give sufficient conditions for solutions to exist without necessarily being unique, which can allow for the existence of singular solutions.".
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- Singular_solution wikiPageLength "7065".
- Singular_solution wikiPageOutDegree "20".
- Singular_solution wikiPageRevisionID "580577244".
- Singular_solution wikiPageWikiLink 0_(number).
- Singular_solution wikiPageWikiLink Algebra.
- Singular_solution wikiPageWikiLink Category:Differential_equations.
- Singular_solution wikiPageWikiLink Caustic_(mathematics).
- Singular_solution wikiPageWikiLink Clairauts_equation.
- Singular_solution wikiPageWikiLink Derivative.
- Singular_solution wikiPageWikiLink Differential_equation.
- Singular_solution wikiPageWikiLink Envelope_(mathematics).
- Singular_solution wikiPageWikiLink Initial_value_problem.
- Singular_solution wikiPageWikiLink Mathematical_singularity.
- Singular_solution wikiPageWikiLink Ordinary_differential_equation.
- Singular_solution wikiPageWikiLink Peano_existence_theorem.
- Singular_solution wikiPageWikiLink Picard–Lindelöf_theorem.
- Singular_solution wikiPageWikiLink Singularity_(mathematics).
- Singular_solution wikiPageWikiLink Tangent.
- Singular_solution wikiPageWikiLinkText "Singular solution".
- Singular_solution wikiPageWikiLinkText "singular solution".
- Singular_solution first "N.Kh.".
- Singular_solution hasPhotoCollection Singular_solution.
- Singular_solution id "Singular_solution".
- Singular_solution last "Rozov".
- Singular_solution oldid "14548".
- Singular_solution wikiPageUsesTemplate Template:SpringerEOM.
- Singular_solution subject Category:Differential_equations.
- Singular_solution hypernym Solution.
- Singular_solution type Software.
- Singular_solution comment "A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full real line.".
- Singular_solution label "Singular solution".
- Singular_solution sameAs Singulární_řešení.
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- Singular_solution sameAs Особое_решение.
- Singular_solution sameAs Q4338245.
- Singular_solution sameAs Q4338245.
- Singular_solution wasDerivedFrom Singular_solution?oldid=580577244.
- Singular_solution isPrimaryTopicOf Singular_solution.