Department of Mathematics and Computer Science
Master of Science
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.
Palmer, Robert, "Dynamics and Asymptotic Behavior of the Solutions of a Nonlinear Differential Equation" (1999). Masters Theses & Specialist Projects. Paper 737.