{"id":{"repo_id":"wku-diss","oai_identifier":"oai:digitalcommons.wku.edu:theses-1740"},"canonical_url":"https://search.dev.ndltd.org/etd/wku-diss/oai:digitalcommons.wku.edu:theses-1740","repository":{"repo_id":"wku-diss","name":"Western Kentucky University","base_url":"https://digitalcommons.wku.edu/do/oai/"},"display":{"title":"Dynamics and Asymptotic Behavior of the Solutions of a Nonlinear Differential Equation","abstract":"Initial value problems of the form dx/dt= t xP, x(a) = β are examined, first when p = 2. Applying Euler's method, a numerical approximation technique, when p = 2 for certain initial conditions produces a numerical solution which resembles a bifurcation diagram very similar to that produced by the logistic map. Comparisons of such numerical solutions to the logistic map are made, and a partial explanation of such numerical solutions is given. Then, the exact solution of the initial value problem with p = 2, for which the software package Mathematica 3.0 determines an explicit formula, is analyzed to determine its uniqueness, range of existence, and dependency upon initial conditions. The long - term behavior of the solution is also determined. Solutions of the initial value problem are also analyzed when p is an integer greater than 2. Conclusions about the behavior of solutions to such initial value problems are made, and such conclusions depend in part upon whether p is even or odd. Mathematica Version 3.0 was unable to determine formulas for selected problems of this form.","abstract_html":"Initial value problems of the form dx/dt= t xP, x(a) = β are examined, first when p = 2. Applying Euler's method, a numerical approximation technique, when p = 2 for certain initial conditions produces a numerical solution which resembles a bifurcation diagram very similar to that produced by the logistic map. Comparisons of such numerical solutions to the logistic map are made, and a partial explanation of such numerical solutions is given. Then, the exact solution of the initial value problem with p = 2, for which the software package Mathematica 3.0 determines an explicit formula, is analyzed to determine its uniqueness, range of existence, and dependency upon initial conditions. The long - term behavior of the solution is also determined. Solutions of the initial value problem are also analyzed when p is an integer greater than 2. Conclusions about the behavior of solutions to such initial value problems are made, and such conclusions depend in part upon whether p is even or odd. Mathematica Version 3.0 was unable to determine formulas for selected problems of this form.","abstract_has_math":false,"creators":["Palmer, Robert"],"institution":null,"degree_name":"Master of Science","degree_level":null,"degree_discipline":"Department of Mathematics and Computer Science","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1999,"date_issued":"1999-08-01T07:00:00Z","date_published":"1999-08-01T07:00:00Z","updated_at":"2026-07-24T06:07:48Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wku.edu/theses/737","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Palmer, Robert"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematics and Computer Science"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.wku.edu/theses/737"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Initial value problems of the form dx/dt= t xP, x(a) = β are examined, first when p = 2. Applying Euler's method, a numerical approximation technique, when p = 2 for certain initial conditions produces a numerical solution which resembles a bifurcation diagram very similar to that produced by the logistic map. Comparisons of such numerical solutions to the logistic map are made, and a partial explanation of such numerical solutions is given. Then, the exact solution of the initial value problem with p = 2, for which the software package Mathematica 3.0 determines an explicit formula, is analyzed to determine its uniqueness, range of existence, and dependency upon initial conditions. The long - term behavior of the solution is also determined. Solutions of the initial value problem are also analyzed when p is an integer greater than 2. Conclusions about the behavior of solutions to such initial value problems are made, and such conclusions depend in part upon whether p is even or odd. Mathematica Version 3.0 was unable to determine formulas for selected problems of this form."]},{"key":"dc:title","label":"Title","values":["Dynamics and Asymptotic Behavior of the Solutions of a Nonlinear Differential Equation"]}]}],"canonical_facts":{"dc:creator":["Palmer, Robert"],"dc:description.abstract":["Initial value problems of the form dx/dt= t xP, x(a) = β are examined, first when p = 2. Applying Euler's method, a numerical approximation technique, when p = 2 for certain initial conditions produces a numerical solution which resembles a bifurcation diagram very similar to that produced by the logistic map. Comparisons of such numerical solutions to the logistic map are made, and a partial explanation of such numerical solutions is given. Then, the exact solution of the initial value problem with p = 2, for which the software package Mathematica 3.0 determines an explicit formula, is analyzed to determine its uniqueness, range of existence, and dependency upon initial conditions. The long - term behavior of the solution is also determined. Solutions of the initial value problem are also analyzed when p is an integer greater than 2. Conclusions about the behavior of solutions to such initial value problems are made, and such conclusions depend in part upon whether p is even or odd. Mathematica Version 3.0 was unable to determine formulas for selected problems of this form."],"dc:identifier":["https://digitalcommons.wku.edu/theses/737"],"dc:subject":["Mathematics"],"dc:title":["Dynamics and Asymptotic Behavior of the Solutions of a Nonlinear Differential Equation"],"dc:type":["Thesis"],"thesis:degree_discipline":["Department of Mathematics and Computer Science"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T06:07:48Z"}