Chapter 2.1, Problem 4E

Interpretation Introduction

Interpretation:

To show that the exact solution of $\dot{\text{x}}\text{ = sin x}$ for ${\text{x}}_{\text{0}}\text{=}\frac{\text{π}}{\text{4}}$ is ${\text{x(t) = 2 tan}}^{\text{-1}}\left(\frac{{\text{e}}^{\text{t}}}{\text{1+}\sqrt{\text{2}}}\right)$ and to conclude that as $\text{t}\to \infty$, $\text{x(t)}\to \text{π}$.

The analytical solution of $\text{x(t)}$ for the given arbitrary initial condition ${\text{x}}_{\text{0}}$ is to be found.

Concept Introduction:

By applyingdifferent trigonometric properties on the given solution of the equation $\dot{\text{x}}\text{ = sin x}$, the solution in terms of ${\text{x}}_{\text{0}}$ can be obtained.

Substitute the initial condition ${\text{x}}_{\text{0}}\text{=}\frac{\text{π}}{\text{4}}$ to get the equation ${\text{x(t) = 2 tan}}^{\text{-1}}\left(\frac{{\text{e}}^{\text{t}}}{\text{1+}\sqrt{\text{2}}}\right)$.

Substitute $\text{t = }\infty$ to conclude that as $\text{t}\to \infty$, $\text{x(t)}\to \text{π}$.

Using the solution for $\text{x(t)}$, the analytical solution in terms of the arbitrary initial condition ${\text{x}}_{\text{0}}$ can be found.