Qualitative theory of differential equations
Identifieur interne : 001856 ( Istex/Corpus ); précédent : 001855; suivant : 001857Qualitative theory of differential equations
Auteurs : Martin BraunSource :
- Applied Mathematical Sciences [ 0066-5452 ]
Abstract
Abstract: In this chapter we consider the differential equation (1) $$ \mathop{x}\limits^{\bullet } = {\text{f}}(t,{\text{x}}) $$ where $$ {\text{x = }}\left[ {\begin{array}{*{20}{c}} {{x_{1}}(t)} \\ \vdots \\ {{x_{n}}(t)} \\ \end{array} } \right], $$ and $$ {\text{f} ={t,x} }\left[ {\begin{array}{*{20}{c}} {{f_{1}}(t,{x_{1}},...,{x_{n}})} \\ \vdots \\ {{f_{n}}(t,{x_{1}},...,{x_{n}})} \\ \end{array} } \right] $$ is a nonlinear function of x 1,...,x n . Unfortunately, there are no known methods of solving Equation (1). This, of course, is very disappointing. However, it is not necessary, in most applications, to find the solutions of (1) explicitly. For example, let x 1(t) and x 2(t) denote the populations, at time t, of two species competing amongst themselves for the limited food and living space in their microcosm. Suppose, moreover, that the rates of growth of x 1(t) and x 2(t) are governed by the differential equation (1). In this case, we are not really interested in the values of x 1(t) and x 2(t) at every time r. Rather, we are interested in the qualitative properties of x 1(t) and x 2(t). Specically, we wish to answer the following questions.
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DOI: 10.1007/978-1-4684-9229-3_4
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Qualitative theory of differential equations</title><author><name sortKey="Braun, Martin" sort="Braun, Martin" uniqKey="Braun M" first="Martin" last="Braun">Martin Braun</name><affiliation><mods:affiliation>Department of Mathematics, Queens College, City University of New York, 11367, Flushing, NY, USA</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:0DB3485EE32AE7441F052EAC93FF62F8A13FCA50</idno><date when="1983" year="1983">1983</date><idno type="doi">10.1007/978-1-4684-9229-3_4</idno><idno type="url">https://api.istex.fr/ark:/67375/HCB-QPTP01WR-C/fulltext.pdf</idno><idno type="wicri:Area/Istex/Corpus">001856</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">001856</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Qualitative theory of differential equations</title><author><name sortKey="Braun, Martin" sort="Braun, Martin" uniqKey="Braun M" first="Martin" last="Braun">Martin Braun</name><affiliation><mods:affiliation>Department of Mathematics, Queens College, City University of New York, 11367, Flushing, NY, USA</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="s" type="main" xml:lang="en">Applied Mathematical Sciences</title><idno type="ISSN">0066-5452</idno><idno type="ISSN">0066-5452</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0066-5452</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: In this chapter we consider the differential equation (1) $$ \mathop{x}\limits^{\bullet } = {\text{f}}(t,{\text{x}}) $$ where $$ {\text{x = }}\left[ {\begin{array}{*{20}{c}} {{x_{1}}(t)} \\ \vdots \\ {{x_{n}}(t)} \\ \end{array} } \right], $$ and $$ {\text{f} ={t,x} }\left[ {\begin{array}{*{20}{c}} {{f_{1}}(t,{x_{1}},...,{x_{n}})} \\ \vdots \\ {{f_{n}}(t,{x_{1}},...,{x_{n}})} \\ \end{array} } \right] $$ is a nonlinear function of x 1,...,x n . Unfortunately, there are no known methods of solving Equation (1). This, of course, is very disappointing. However, it is not necessary, in most applications, to find the solutions of (1) explicitly. For example, let x 1(t) and x 2(t) denote the populations, at time t, of two species competing amongst themselves for the limited food and living space in their microcosm. Suppose, moreover, that the rates of growth of x 1(t) and x 2(t) are governed by the differential equation (1). In this case, we are not really interested in the values of x 1(t) and x 2(t) at every time r. Rather, we are interested in the qualitative properties of x 1(t) and x 2(t). Specically, we wish to answer the following questions.</div></front></TEI><istex><corpusName>springer-ebooks</corpusName><author><json:item><name>Martin Braun</name><affiliations><json:string>Department of Mathematics, Queens College, City University of New York, 11367, Flushing, NY, USA</json:string></affiliations></json:item></author><arkIstex>ark:/67375/HCB-QPTP01WR-C</arkIstex><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><abstract>Abstract: In this chapter we consider the differential equation (1) $$ \mathop{x}\limits^{\bullet } = {\text{f}}(t,{\text{x}}) $$ where $$ {\text{x = }}\left[ {\begin{array}{*{20}{c}} {{x_{1}}(t)} \\ \vdots \\ {{x_{n}}(t)} \\ \end{array} } \right], $$ and $$ {\text{f} ={t,x} }\left[ {\begin{array}{*{20}{c}} {{f_{1}}(t,{x_{1}},...,{x_{n}})} \\ \vdots \\ {{f_{n}}(t,{x_{1}},...,{x_{n}})} \\ \end{array} } \right] $$ is a nonlinear function of x 1,...,x n . Unfortunately, there are no known methods of solving Equation (1). This, of course, is very disappointing. However, it is not necessary, in most applications, to find the solutions of (1) explicitly. For example, let x 1(t) and x 2(t) denote the populations, at time t, of two species competing amongst themselves for the limited food and living space in their microcosm. Suppose, moreover, that the rates of growth of x 1(t) and x 2(t) are governed by the differential equation (1). In this case, we are not really interested in the values of x 1(t) and x 2(t) at every time r. Rather, we are interested in the qualitative properties of x 1(t) and x 2(t). Specically, we wish to answer the following questions.</abstract><qualityIndicators><score>9.172</score><pdfWordCount>30573</pdfWordCount><pdfCharCount>153564</pdfCharCount><pdfVersion>1.4</pdfVersion><pdfPageCount>104</pdfPageCount><pdfPageSize>439 x 666 pts</pdfPageSize><refBibsNative>false</refBibsNative><abstractWordCount>181</abstractWordCount><abstractCharCount>1168</abstractCharCount><keywordCount>0</keywordCount></qualityIndicators><title>Qualitative theory of differential equations</title><chapterId><json:string>4</json:string><json:string>Chap4</json:string></chapterId><genre><json:string>research-article</json:string></genre><serie><title>Applied Mathematical Sciences</title><language><json:string>unknown</json:string></language><copyrightDate>1983</copyrightDate><issn><json:string>0066-5452</json:string></issn><editor><json:item><name>F. John</name><affiliations><json:string>Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA</json:string></affiliations></json:item><json:item><name>J. E. Marsden</name><affiliations><json:string>Department of Mathematics, University of California, 94720, Berkelay, CA, USA</json:string></affiliations></json:item><json:item><name>L. Sirovich</name><affiliations><json:string>Division of Applied Mathematics, Brown University, 02912, Providence, RI, USA</json:string></affiliations></json:item></editor></serie><host><title>Differential Equations and Their Applications</title><language><json:string>unknown</json:string></language><copyrightDate>1983</copyrightDate><doi><json:string>10.1007/978-1-4684-9229-3</json:string></doi><issn><json:string>0066-5452</json:string></issn><eisbn><json:string>978-1-4684-9229-3</json:string></eisbn><bookId><json:string>978-1-4684-9229-3</json:string></bookId><isbn><json:string>978-0-387-97938-0</json:string></isbn><volume>15</volume><pages><first>370</first><last>473</last></pages><genre><json:string>book-series</json:string></genre><author><json:item><name>Martin Braun</name><affiliations><json:string>Department of Mathematics, Queens College, City University of New York, 11367, Flushing, NY, USA</json:string></affiliations></json:item></author><subject><json:item><value>Mathematics and Statistics</value></json:item><json:item><value>Mathematics</value></json:item><json:item><value>Analysis</value></json:item><json:item><value>Numerical and Computational Physics</value></json:item></subject></host><ark><json:string>ark:/67375/HCB-QPTP01WR-C</json:string></ark><publicationDate>1983</publicationDate><copyrightDate>1983</copyrightDate><doi><json:string>10.1007/978-1-4684-9229-3_4</json:string></doi><id>0DB3485EE32AE7441F052EAC93FF62F8A13FCA50</id><score>1</score><fulltext><json:item><extension>pdf</extension><original>true</original><mimetype>application/pdf</mimetype><uri>https://api.istex.fr/ark:/67375/HCB-QPTP01WR-C/fulltext.pdf</uri></json:item><json:item><extension>zip</extension><original>false</original><mimetype>application/zip</mimetype><uri>https://api.istex.fr/ark:/67375/HCB-QPTP01WR-C/bundle.zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/ark:/67375/HCB-QPTP01WR-C/fulltext.tei"><teiHeader><fileDesc><titleStmt><title level="a" type="main" xml:lang="en">Qualitative theory of differential equations</title></titleStmt><publicationStmt><authority>ISTEX</authority><availability><licence>Springer-Verlag New York, Inc.</licence></availability><date when="1983">1983</date></publicationStmt><notesStmt><note type="content-type" subtype="research-article" source="OriginalPaper" scheme="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</note><note type="publication-type" subtype="book-series" scheme="https://publication-type.data.istex.fr/ark:/67375/JMC-0G6R5W5T-Z">book-series</note></notesStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Qualitative theory of differential equations</title><author><persName><forename type="first">Martin</forename><surname>Braun</surname></persName><affiliation><orgName type="department">Department of Mathematics, Queens College</orgName><orgName type="institution">City University of New York</orgName><address><settlement>Flushing</settlement><region>NY</region><postCode>11367</postCode><country key="US">UNITED STATES</country></address></affiliation></author><idno type="istex">0DB3485EE32AE7441F052EAC93FF62F8A13FCA50</idno><idno type="ark">ark:/67375/HCB-QPTP01WR-C</idno><idno type="DOI">10.1007/978-1-4684-9229-3_4</idno></analytic><monogr><title level="m" type="main">Differential Equations and Their Applications</title><title level="m" type="sub">An Introduction to Applied Mathematics</title><idno type="DOI">10.1007/978-1-4684-9229-3</idno><idno type="book-id">978-1-4684-9229-3</idno><idno type="ISBN">978-0-387-97938-0</idno><idno type="eISBN">978-1-4684-9229-3</idno><idno type="chapter-id">Chap4</idno><author><persName><forename type="first">Martin</forename><surname>Braun</surname></persName><affiliation><orgName type="department">Department of Mathematics, Queens College</orgName><orgName type="institution">City University of New York</orgName><address><settlement>Flushing</settlement><region>NY</region><postCode>11367</postCode><country key="US">UNITED STATES</country></address></affiliation></author><imprint><biblScope unit="vol">15</biblScope><biblScope unit="page" from="370">370</biblScope><biblScope unit="page" to="473">473</biblScope><biblScope unit="chapter-count">5</biblScope></imprint></monogr><series><title level="s" type="main" xml:lang="en">Applied Mathematical Sciences</title><editor><persName><forename type="first">F.</forename><surname>John</surname></persName><affiliation><orgName type="department">Courant Institute of Mathematical Sciences</orgName><orgName type="institution">New York University</orgName><address><settlement>New York</settlement><region>NY</region><postCode>10012</postCode><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">J.</forename><forename type="first">E.</forename><surname>Marsden</surname></persName><affiliation><orgName type="department">Department of Mathematics</orgName><orgName type="institution">University of California</orgName><address><settlement>Berkelay</settlement><region>CA</region><postCode>94720</postCode><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">L.</forename><surname>Sirovich</surname></persName><affiliation><orgName type="department">Division of Applied Mathematics</orgName><orgName type="institution">Brown University</orgName><address><settlement>Providence</settlement><region>RI</region><postCode>02912</postCode><country key="US">UNITED STATES</country></address></affiliation></editor><idno type="pISSN">0066-5452</idno><idno type="seriesID">34</idno></series></biblStruct></sourceDesc></fileDesc><profileDesc><abstract xml:lang="en"><head>Abstract</head><p>In this chapter we consider the differential equation <formula xml:id="Equ1" notation="TEX" n="(1)"><media mimeType="image" url=""></media>
$$ \mathop{x}\limits^{\bullet } = {\text{f}}(t,{\text{x}}) $$
</formula> where <formula xml:id="Equ2" notation="TEX"><media mimeType="image" url=""></media>
$$ {\text{x = }}\left[ {\begin{array}{*{20}{c}} {{x_{1}}(t)} \\ \vdots \\ {{x_{n}}(t)} \\ \end{array} } \right], $$
</formula> and <formula xml:id="Equ3" notation="TEX"><media mimeType="image" url=""></media>
$$ {\text{f} ={t,x} }\left[ {\begin{array}{*{20}{c}} {{f_{1}}(t,{x_{1}},...,{x_{n}})} \\ \vdots \\ {{f_{n}}(t,{x_{1}},...,{x_{n}})} \\ \end{array} } \right] $$
</formula> is a nonlinear function of <hi rend="italic">x</hi><hi rend="subscript">1</hi>,...,<hi rend="italic">x</hi><hi rend="subscript"><hi rend="italic">n</hi></hi>. Unfortunately, there are no known methods of solving Equation (1). This, of course, is very disappointing. However, it is not necessary, in most applications, to find the solutions of (1) explicitly. For example, let <hi rend="italic">x</hi><hi rend="subscript">1</hi>(<hi rend="italic">t</hi>) and <hi rend="italic">x</hi><hi rend="subscript">2</hi>(<hi rend="italic">t</hi>) denote the populations, at time <hi rend="italic">t</hi>, of two species competing amongst themselves for the limited food and living space in their microcosm. Suppose, moreover, that the rates of growth of <hi rend="italic">x</hi><hi rend="subscript">1</hi>(<hi rend="italic">t</hi>) and <hi rend="italic">x</hi><hi rend="subscript">2</hi>(<hi rend="italic">t</hi>) are governed by the differential equation (1). In this case, we are not really interested in the values of <hi rend="italic">x</hi><hi rend="subscript">1</hi>(<hi rend="italic">t</hi>) and <hi rend="italic">x</hi><hi rend="subscript">2</hi>(<hi rend="italic">t</hi>) at every time r. Rather, we are interested in the qualitative properties of <hi rend="italic">x</hi><hi rend="subscript">1</hi>(<hi rend="italic">t</hi>) and <hi rend="italic">x</hi><hi rend="subscript">2</hi>(<hi rend="italic">t</hi>). Specically, we wish to answer the following questions.</p></abstract><textClass ana="subject"><keywords scheme="book-subject-collection"><list><label>SUCO11649</label><item><term>Mathematics and Statistics</term></item></list></keywords></textClass><textClass ana="subject"><keywords scheme="book-subject"><list><label>SCM</label><item><term type="Primary">Mathematics</term></item><label>SCM12007</label><item><term type="Secondary" subtype="priority-1">Analysis</term></item><label>SCP19021</label><item><term type="Secondary" subtype="priority-2">Numerical and Computational Physics</term></item></list></keywords></textClass><langUsage><language ident="EN"></language></langUsage></profileDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/ark:/67375/HCB-QPTP01WR-C/fulltext.txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="corpus springer-ebooks not found" 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DisplayOrder="Western"><GivenName>J.</GivenName><GivenName>E.</GivenName><FamilyName>Marsden</FamilyName></EditorName></Editor><Editor AffiliationIDS="Aff3"><EditorName DisplayOrder="Western"><GivenName>L.</GivenName><FamilyName>Sirovich</FamilyName></EditorName></Editor><Affiliation ID="Aff1"><OrgDivision>Courant Institute of Mathematical Sciences</OrgDivision><OrgName>New York University</OrgName><OrgAddress><City>New York</City><State>NY</State><Postcode>10012</Postcode><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgDivision>Department of Mathematics</OrgDivision><OrgName>University of California</OrgName><OrgAddress><City>Berkelay</City><State>CA</State><Postcode>94720</Postcode><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff3"><OrgDivision>Division of Applied Mathematics</OrgDivision><OrgName>Brown University</OrgName><OrgAddress><City>Providence</City><State>RI</State><Postcode>02912</Postcode><Country>USA</Country></OrgAddress></Affiliation></EditorGroup></SeriesHeader><Book Language="En"><BookInfo BookProductType="Undergraduate textbook" ContainsESM="No" Language="En" MediaType="eBook" NumberingStyle="ChapterOnly" OutputMedium="All" TocLevels="0"><BookID>978-1-4684-9229-3</BookID><BookTitle>Differential Equations and Their Applications</BookTitle><BookSubTitle>An Introduction to Applied Mathematics</BookSubTitle><BookVolumeNumber>15</BookVolumeNumber><BookSequenceNumber>15</BookSequenceNumber><BookDOI>10.1007/978-1-4684-9229-3</BookDOI><BookTitleID>17766</BookTitleID><BookPrintISBN>978-0-387-97938-0</BookPrintISBN><BookElectronicISBN>978-1-4684-9229-3</BookElectronicISBN><EditionNumber>3</EditionNumber><BookEdition>3rd Edition</BookEdition><BookChapterCount>5</BookChapterCount><BookCopyright><CopyrightHolderName>Springer-Verlag New York</CopyrightHolderName><CopyrightYear>1983</CopyrightYear></BookCopyright><BookSubjectGroup><BookSubject Code="SCM" Type="Primary">Mathematics</BookSubject><BookSubject Code="SCM12007" Priority="1" Type="Secondary">Analysis</BookSubject><BookSubject Code="SCP19021" Priority="2" Type="Secondary">Numerical and Computational Physics</BookSubject><SubjectCollection Code="SUCO11649">Mathematics and Statistics</SubjectCollection></BookSubjectGroup><BookContext><SeriesID>34</SeriesID></BookContext></BookInfo><BookHeader><AuthorGroup><Author AffiliationIDS="Aff4"><AuthorName DisplayOrder="Western"><GivenName>Martin</GivenName><FamilyName>Braun</FamilyName></AuthorName></Author><Affiliation ID="Aff4"><OrgDivision>Department of Mathematics, Queens College</OrgDivision><OrgName>City University of New York</OrgName><OrgAddress><City>Flushing</City><State>NY</State><Postcode>11367</Postcode><Country>USA</Country></OrgAddress></Affiliation></AuthorGroup></BookHeader><Chapter ID="Chap4" Language="En"><ChapterInfo ChapterType="OriginalPaper" ContainsESM="No" Language="En" NumberingStyle="ChapterOnly" OutputMedium="All" TocLevels="0"><ChapterID>4</ChapterID><ChapterNumber>4</ChapterNumber><ChapterDOI>10.1007/978-1-4684-9229-3_4</ChapterDOI><ChapterSequenceNumber>4</ChapterSequenceNumber><ChapterTitle Language="En">Qualitative theory of differential equations</ChapterTitle><ChapterFirstPage>370</ChapterFirstPage><ChapterLastPage>473</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag New York, Inc.</CopyrightHolderName><CopyrightYear>1983</CopyrightYear></ChapterCopyright><ChapterHistory><RegistrationDate><Year>2012</Year><Month>4</Month><Day>12</Day></RegistrationDate></ChapterHistory><ChapterGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ChapterGrants><ChapterContext><SeriesID>34</SeriesID><BookID>978-1-4684-9229-3</BookID><BookTitle>Differential Equations and Their Applications</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff5"><AuthorName DisplayOrder="Western"><GivenName>Martin</GivenName><FamilyName>Braun</FamilyName></AuthorName></Author><Affiliation ID="Aff5"><OrgDivision>Department of Mathematics, Queens College</OrgDivision><OrgName>City University of New York</OrgName><OrgAddress><City>Flushing</City><State>NY</State><Postcode>11367</Postcode><Country>USA</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En" OutputMedium="Online"><Heading>Abstract</Heading><Para TextBreak="No">In this chapter we consider the differential equation <Equation ID="Equ1"><EquationNumber>(1)</EquationNumber><MediaObject><ImageObject Color="BlackWhite" FileRef="978-1-4684-9229-3_4_Chapter_TeX2GIF_Equ1.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></MediaObject><EquationSource Format="TEX">$$ \mathop{x}\limits^{\bullet } = {\text{f}}(t,{\text{x}}) $$</EquationSource></Equation> where <Equation ID="Equ2"><MediaObject><ImageObject Color="BlackWhite" FileRef="978-1-4684-9229-3_4_Chapter_TeX2GIF_Equ2.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></MediaObject><EquationSource Format="TEX">$$ {\text{x = }}\left[ {\begin{array}{*{20}{c}} {{x_{1}}(t)} \\ \vdots \\ {{x_{n}}(t)} \\ \end{array} } \right], $$</EquationSource></Equation> and <Equation ID="Equ3"><MediaObject><ImageObject Color="BlackWhite" FileRef="978-1-4684-9229-3_4_Chapter_TeX2GIF_Equ3.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></MediaObject><EquationSource Format="TEX">$$ {\text{f} ={t,x} }\left[ {\begin{array}{*{20}{c}} {{f_{1}}(t,{x_{1}},...,{x_{n}})} \\ \vdots \\ {{f_{n}}(t,{x_{1}},...,{x_{n}})} \\ \end{array} } \right] $$</EquationSource></Equation> is a nonlinear function of <Emphasis Type="Italic">x</Emphasis><Subscript>1</Subscript>,...,<Emphasis Type="Italic">x</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>. Unfortunately, there are no known methods of solving Equation (1). This, of course, is very disappointing. However, it is not necessary, in most applications, to find the solutions of (1) explicitly. For example, let <Emphasis Type="Italic">x</Emphasis><Subscript>1</Subscript>(<Emphasis Type="Italic">t</Emphasis>) and <Emphasis Type="Italic">x</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">t</Emphasis>) denote the populations, at time <Emphasis Type="Italic">t</Emphasis>, of two species competing amongst themselves for the limited food and living space in their microcosm. Suppose, moreover, that the rates of growth of <Emphasis Type="Italic">x</Emphasis><Subscript>1</Subscript>(<Emphasis Type="Italic">t</Emphasis>) and <Emphasis Type="Italic">x</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">t</Emphasis>) are governed by the differential equation (1). In this case, we are not really interested in the values of <Emphasis Type="Italic">x</Emphasis><Subscript>1</Subscript>(<Emphasis Type="Italic">t</Emphasis>) and <Emphasis Type="Italic">x</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">t</Emphasis>) at every time r. Rather, we are interested in the qualitative properties of <Emphasis Type="Italic">x</Emphasis><Subscript>1</Subscript>(<Emphasis Type="Italic">t</Emphasis>) and <Emphasis Type="Italic">x</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">t</Emphasis>). Specically, we wish to answer the following questions.</Para></Abstract></ChapterHeader><NoBody></NoBody></Chapter></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Qualitative theory of differential equations</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>Qualitative theory of differential equations</title></titleInfo><name type="personal"><namePart type="given">Martin</namePart><namePart type="family">Braun</namePart><affiliation>Department of Mathematics, Queens College, City University of New York, 11367, Flushing, NY, USA</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" type="research-article" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Springer New York</publisher><place><placeTerm type="text">New York, NY</placeTerm></place><dateIssued encoding="w3cdtf">1983</dateIssued><copyrightDate encoding="w3cdtf">1983</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: In this chapter we consider the differential equation (1) $$ \mathop{x}\limits^{\bullet } = {\text{f}}(t,{\text{x}}) $$ where $$ {\text{x = }}\left[ {\begin{array}{*{20}{c}} {{x_{1}}(t)} \\ \vdots \\ {{x_{n}}(t)} \\ \end{array} } \right], $$ and $$ {\text{f} ={t,x} }\left[ {\begin{array}{*{20}{c}} {{f_{1}}(t,{x_{1}},...,{x_{n}})} \\ \vdots \\ {{f_{n}}(t,{x_{1}},...,{x_{n}})} \\ \end{array} } \right] $$ is a nonlinear function of x 1,...,x n . Unfortunately, there are no known methods of solving Equation (1). This, of course, is very disappointing. However, it is not necessary, in most applications, to find the solutions of (1) explicitly. For example, let x 1(t) and x 2(t) denote the populations, at time t, of two species competing amongst themselves for the limited food and living space in their microcosm. Suppose, moreover, that the rates of growth of x 1(t) and x 2(t) are governed by the differential equation (1). In this case, we are not really interested in the values of x 1(t) and x 2(t) at every time r. Rather, we are interested in the qualitative properties of x 1(t) and x 2(t). Specically, we wish to answer the following questions.</abstract><relatedItem type="host"><titleInfo><title>Differential Equations and Their Applications</title><subTitle>An Introduction to Applied Mathematics</subTitle></titleInfo><name type="personal"><namePart type="given">Martin</namePart><namePart type="family">Braun</namePart><affiliation>Department of Mathematics, Queens College, City University of New York, 11367, Flushing, NY, USA</affiliation></name><genre type="book-series" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0G6R5W5T-Z">book-series</genre><originInfo><publisher>Springer</publisher><copyrightDate encoding="w3cdtf">1983</copyrightDate><issuance>monographic</issuance><edition>3rd Edition</edition></originInfo><subject><genre>Book-Subject-Collection</genre><topic authority="SpringerSubjectCodes" authorityURI="SUCO11649">Mathematics and Statistics</topic></subject><subject><genre>Book-Subject-Group</genre><topic authority="SpringerSubjectCodes" authorityURI="SCM">Mathematics</topic><topic authority="SpringerSubjectCodes" authorityURI="SCM12007">Analysis</topic><topic authority="SpringerSubjectCodes" authorityURI="SCP19021">Numerical and Computational Physics</topic></subject><identifier type="DOI">10.1007/978-1-4684-9229-3</identifier><identifier type="ISBN">978-0-387-97938-0</identifier><identifier type="eISBN">978-1-4684-9229-3</identifier><identifier type="ISSN">0066-5452</identifier><identifier type="BookTitleID">17766</identifier><identifier type="BookID">978-1-4684-9229-3</identifier><identifier type="BookChapterCount">5</identifier><identifier type="BookVolumeNumber">15</identifier><identifier type="BookSequenceNumber">15</identifier><part><date>1983</date><detail type="volume"><number>15</number><caption>vol.</caption></detail><detail type="chapter"><number>4</number><caption>chapter</caption></detail><extent unit="pages"><start>370</start><end>473</end></extent></part><recordInfo><recordOrigin>Springer-Verlag New York, 1983</recordOrigin></recordInfo></relatedItem><relatedItem type="series"><titleInfo><title>Applied Mathematical Sciences</title></titleInfo><name type="personal"><namePart type="given">F.</namePart><namePart type="family">John</namePart><affiliation>Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">J.</namePart><namePart type="given">E.</namePart><namePart type="family">Marsden</namePart><affiliation>Department of Mathematics, University of California, 94720, Berkelay, CA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">L.</namePart><namePart type="family">Sirovich</namePart><affiliation>Division of Applied Mathematics, Brown University, 02912, Providence, RI, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><originInfo><publisher>Springer</publisher><copyrightDate encoding="w3cdtf">1983</copyrightDate><issuance>serial</issuance></originInfo><identifier type="ISSN">0066-5452</identifier><identifier type="SeriesID">34</identifier><part><detail type="chapter"><number>4</number></detail></part><recordInfo><recordOrigin>Springer-Verlag New York, 1983</recordOrigin></recordInfo></relatedItem><identifier type="istex">0DB3485EE32AE7441F052EAC93FF62F8A13FCA50</identifier><identifier type="ark">ark:/67375/HCB-QPTP01WR-C</identifier><identifier type="DOI">10.1007/978-1-4684-9229-3_4</identifier><identifier type="ChapterID">4</identifier><identifier type="ChapterID">Chap4</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer-Verlag New York, 1983</accessCondition><recordInfo><recordContentSource authority="ISTEX" authorityURI="https://loaded-corpus.data.istex.fr" valueURI="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-RLRX46XW-4">springer</recordContentSource><recordOrigin>Springer-Verlag New York, Inc., 1983</recordOrigin></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/ark:/67375/HCB-QPTP01WR-C/record.json</uri></json:item></metadata><annexes><json:item><extension>xml</extension><original>true</original><mimetype>application/xml</mimetype><uri>https://api.istex.fr/ark:/67375/HCB-QPTP01WR-C/annexes.xml</uri></json:item></annexes></istex></record>Pour manipuler ce document sous Unix (Dilib)
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