Chapter 2, Problem 2.1P

(a)

To determine

Check whether the given function is integrable.

Answer to Problem 2.1P

The given function is not integrable.

Explanation of Solution

The basic equation is

    $F=m_i{\ddot{x}}_i$

The given equation is

    $\begin{array}{c}F\left(x_i,t\right)=f\left(x_i\right)g\left(t\right)\\ =m_i{\ddot{x}}_i\end{array}$

This is not integrable.

Conclusion:

The given function is not integrable.

(b)

To determine

Check whether the given function is integrable.

Answer to Problem 2.1P

The given function is integrable.

Explanation of Solution

The given function is

    $\begin{array}{c}F\left({\dot{x}}_i,t\right)=f\left({\dot{x}}_i\right)g\left(t\right)\\ =m_i{\ddot{x}}_i\\ f\left({\dot{x}}_i\right)g\left(t\right)=m_i\frac{d{\dot{x}}_i}{dt}\\ \frac{d{\dot{x}}_i}{f\left({\dot{x}}_i\right)}=\frac{g\left(t\right)}{m_i}\end{array}$

This is integrable.

Conclusion:

The given function is integrable.

(c)

To determine

Check whether the given function is integrable.

Answer to Problem 2.1P

The given function is not integrable.

Explanation of Solution

The basic equation is

    $F=m_i{\ddot{x}}_i$

The given equation is

    $\begin{array}{c}F\left(x_i,{\dot{x}}_i\right)=f\left(x_i\right)g\left({\dot{x}}_i\right)\\ =m_i{\ddot{x}}_i\end{array}$

This is not integrable.

Conclusion:

The given function is not integrable.