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Differential Equations and the Method of Upper and Lower Solutions
In this paper, some background material regarding differential equations and initial value problems is presented. The method of upper and lower solutions, which is used for determining existence of periodic solutions to periodic differential equations, is then discussed. Theorems regarding periodicity and the first-order case of upper and lower solutions are proven. The method is applied to some examples from pure mathematics along with the logistic equation, and corresponding graphs generated in MATLAB illustrate the periodic behavior and stability of solutions. The second-order case of upper and lower solutions is then introduced, and an example is taken from pure mathematics in addition to one regarding a simple undamped pendulum subject to periodic forcing. In conclusion, it is noted that the method of upper and lower solutions is used for existence purposes only and should not be used if analytical solutions can be obtained; the method somewhat resembles the intermediate value theorem and squeeze theorem; and it is useful mainly for nonlinear periodic differential equations when analytical solutions do not exist.
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