Answered: y = a sech-1 |a² – x², a > 0. a

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Consider the equation of a tractrix (a) Find dy/dx. (b) Let L be the tangent line to the tractrix at the point P. When L intersects the y-axis at the point Q, show that the distance between P and Q is a.

Expert Solution

Step 1

Given: Equation of a tractrix is $y = a s e c h^{- 1} (\frac{x}{a}) - \sqrt{a^{2} - x^{2}}, a > 0$ .

To Find :

a) $\frac{d y}{d x}$

b) When L intersect the $y$ -axis at the point Q show that the distance between P and Q is $a$ .

Step 2

The equation of tractrix is $y = a s e c h^{- 1} (\frac{x}{a}) - \sqrt{a^{2} - x^{2}}, a > 0$ .

Now, differentiate $y$ with respect of $x$ by applying the chain and sum rule.

$\begin{array}{rcl} y & = & a s e c h^{- 1} (\frac{x}{a}) - \sqrt{a^{2} - x^{2}} \\ \frac{d y}{d x} & = & a (- \frac{1}{\frac{x}{a} \sqrt{1 - {(\frac{x}{a})}^{2}}}) - \frac{1}{2 \sqrt{a^{2} - x^{2}}} (\frac{d}{d x} (- x^{2})) \\ = & a (- \frac{1}{x \sqrt{1 - {(\frac{x}{a})}^{2}}}) + \frac{2 x}{2 \sqrt{a^{2} - x^{2}}} \\ = & (- \frac{a^{2}}{x \sqrt{a^{2} - x^{2}}}) + \frac{x}{\sqrt{a^{2} - x^{2}}} \\ = & - (\frac{a^{2} - x^{2}}{x \sqrt{a^{2} - x^{2}}}) \\ = & - \frac{\sqrt{a^{2} - x^{2}}}{x} \end{array}$

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