Chapter 2.5, Problem 1E

Interpretation Introduction

Interpretation:

The values of $\text{c}$ are to be found such that the origin $\text{x = 0}$ is a stable fixed point if a particle travels on the half-line $\text{x}\ge \text{0}$ with a velocity given by ${\dot{\text{x}}\text{ = -x}}^{\text{c}}$, where $\text{c}$ is real and constant. To find how long does it take for the particle to travel from $\text{x = 1}$ to $\text{x = 0}$ if $\text{c}$ is chosen such that $\text{x = 0}$ is stable.

Concept Introduction:

Fixed points are the points where $\dot{\text{x}}\text{ = 0}$.

If $\text{x}$ is the fixed point of the system, then it is stable if $\ddot{\text{x}}\text{ < 0}$, and the flow is toward the left. It is unstable if $\ddot{\text{x}}\text{ > 0}$, and the flow is toward the right.

Time taken by the particle can be found by integrating the given equation and substituting the given conditions.