Chapter 2.6, Problem 2E

Interpretation Introduction

Interpretation:

For equation ${\displaystyle {\int }_{\text{t}}^{\text{t+T}}\text{f(x)}\frac{\text{dx}}{\text{dt}}\text{dt }}$ the periodic solutions are impossible for a vector field on a line is to be proved.

Concept Introduction:

Assume the $\text{x(t)}$ is the periodic solution for the given equation.

By solving the equation we found that periodic solutions are impossible for a vector field on a line.

That means all trajectory are forced to remain constant.

The phase point never reverses the direction due to that periodic solution is impossible for vector field on a line.