Give an example of a two-dimensional nonlinear first order...

Give an example of a two-dimensional nonlinear first order system for which the hypotheses of Theorem $6.1$ are not satisfied at precisely the specified points in $t y_{1} y_{2}$-space.
$$
\text { The points }\left(t, y_{1}, y_{2}\right)=(1, n \pi, 2), n=0, \pm 1, \pm 2, \ldots
$$

Key Concepts

Two-Dimensional Nonlinear First Order System

This concept refers to a system of differential equations comprised of two equations, each involving first derivatives of the unknown functions with respect to the independent variable (often time). Such systems can involve nonlinear terms (functions of the state variables in a nonlinear way), which makes their qualitative behavior more complex than that of linear systems. They are central in the study of dynamical systems and the theory of differential equations.

Existence and Uniqueness Theorem (Local Version)

In the context of ordinary differential equations, existence and uniqueness theorems (such as the Picard–Lindelöf theorem) guarantee that, given a set of conditions on the right?hand side of the system (usually including continuity and a Lipschitz condition in the state variables), there exists a unique solution to the initial value problem locally in time. The theorem typically does not hold if its hypotheses, including the Lipschitz condition, fail at some points.

Lipschitz Condition and Its Failure at Specified Points

A Lipschitz condition is a type of uniform bound on how fast a function can change with respect to its arguments. It requires that the difference in the function’s values between any two points is bounded by a constant multiple of the distance between those points. In many existence and uniqueness theorems, such as Theorem 6.1, the Lipschitz condition is a key hypothesis. In the example under discussion, one can construct a system where the function (for example, a term like ?|sin(y?)|) fails to be Lipschitz precisely at those points where sin(y?) vanishes, i.e. when y? equals an integer multiple of ?, and simultaneously at the specific values of the other variables (t = 1 and y? = 2). This creates a situation where the local uniqueness and/or existence properties break down exactly at the designated points.

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