Verify the hypotheses of the existence and uniqueness theorem. To show x(t) = sin t and 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 x ˙ = f(x), x(0) = x 0 , if 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.

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Chapter 6.2, Problem 2E

Interpretation Introduction

Interpretation:

Verify the hypotheses of the existence and uniqueness theorem. To show x(t) = sin t and 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 x˙ = f(x), x(0) = x0, if 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.

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