Chapter A6, Problem 69E

To determine

To solve: an equation for the curve.

Answer to Problem 69E

An equation for the curve is $y={\sec }^{-1}\left(x\right)-\sqrt{1-x^2}$

Explanation of Solution

Given:

  

Differential equation: $\frac{dy}{dx}=-\frac{1}{x\sqrt{1-x^2}}+\frac{x}{\sqrt{1-x^2}}$

Initial condition: $y=0whenx=1$

Calculation:

The given differential equation for the path, that is, the Tractrix is,

  $\frac{dy}{dx}=\frac{-1}{x\sqrt{1-x^2}}+\frac{-1}{\sqrt{1-x^2}}$

On integrating both sides we get.

  $\begin{array}{l}y={\displaystyle \int \frac{-1}{x\sqrt{1-x^2}}}\cdot dx+{\displaystyle \int \frac{-1}{x\sqrt{1-x^2}}}\cdot dx\\ =\sec h^{-1}\left(x\right)-\sqrt{1-x^2}+c\end{array}$

Now applying the known conditions, that is $x=1$ when $y=0$

Thus,

  $\begin{array}{l}\Rightarrow C=0\\ \Rightarrow \boxed{y={\sec }^{-1}\left(x\right)-\sqrt{1-x^2}}\end{array}$

Conclusion:

An equation for the curve is $y={\sec }^{-1}\left(x\right)-\sqrt{1-x^2}$