Chapter 6.2, Problem 2E

Interpretation Introduction

Interpretation:

Verify the hypotheses of the existence and uniqueness theorem. To show $\text{x(t) = sin t}$ and $\text{y(t) = cos t}$ are the exact solutions of a system.

Concept Introduction:

The parametric curves traced by solutions of a differential equation are known as trajectories.

The geometrical representation of a collection of trajectories in a phase plane is called a phase portrait.

Existence and Uniqueness Theorem: For initial value problem ${\dot{\text{x}}\text{ = f(x), x(0) = x}}_0$, if $\text{f}$ is continuously differentiable, then there always exist a unique solution for the differential equation.

The corollary of Existence and Uniqueness Theorem: Different trajectories never intersect each other. If two trajectories intersect each other, then there is a possibility of the existence of the two solutions for the same point (intersecting point). This means trajectory moves in two directions from the same point which is not possible.