Qualitative theory of differential equations
Identifieur interne : 002747 ( Main/Merge ); précédent : 002746; suivant : 002748Qualitative theory of differential equations
Auteurs : Martin Braun [États-Unis]Source :
- 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></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><idno type="wicri:Area/Istex/Curation">001856</idno><idno type="wicri:Area/Istex/Checkpoint">001271</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Checkpoint">001271</idno><idno type="wicri:doubleKey">0066-5452:1983:Braun M:qualitative:theory:of</idno><idno type="wicri:Area/Main/Merge">002747</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 wicri:level="2"><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Mathematics, Queens College, City University of New York, 11367, Flushing, NY</wicri:regionArea><placeName><region type="state">État de New York</region></placeName></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></record>Pour manipuler ce document sous Unix (Dilib)
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