Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Verify that $x^{2} + {cy}^{2} = 1,$ where c is an arbitrary non-zero constant, is a one-parameter family of implicit solutions to $\frac{dy}{dx} = \frac{xy}{x^{2} - 1}$ and graph several of the solution curves using the same coordinate axes.

On differentiating the given function $x^{2} + c y^{2} = 1$ with respect to x, we will find that the result is identical to the given differential equation. Hence, $x^{2} + c y^{2} = 1$ is a one-parameter family of implicit solutions to $\frac{d y}{d x} = \frac{x y}{x^{2} - 1}$ for c as an arbitrary non-zero constant.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Important formula.

The required formula is $\frac{d}{dx} (x^{n}) = n x^{n - 1}$ .

Step 2: Taking the given function as a function of x in y.

$y = \sqrt{\frac{1 - x^{2}}{c}}$

Step 3: Differentiate the function in step 2, with respect to x.

$\begin{array}{l} \frac{dy}{dx} = \frac{1}{2 \sqrt{c}} {(1 - x^{2})}^{\frac{- 1}{2}} \times (- 2 x) \\ \frac{dy}{dx} = - \frac{x}{\sqrt{c}} \times \frac{1}{\sqrt{1 - x^{2}}} \end{array}$

Step 4: Simplification of the differential equation obtained in step 2.

Multiplying and dividing the final differential equation obtained in Step 2 by $\sqrt{1 - x^{2}}$ :

role="math" localid="1663943533560" $\begin{array}{l} \frac{dy}{dx} = - \frac{x}{\sqrt{c}} \times \frac{1}{\sqrt{1 - x^{2}}} \frac{\sqrt{1 - x^{2}}}{\sqrt{1 - x^{2}}} \\ \frac{dy}{dx} = - \frac{x}{(1 - x^{2})} (\frac{\sqrt{1 - x^{2}}}{\sqrt{c}}) \\ \frac{d y}{d x} = - \frac{x y}{(1 - x^{2})} \\ \frac{d y}{d x} = \frac{x y}{(x^{2} - 1)} \end{array}$

Which is identical to the given differential equation.

Hence, $x^{2} + c y^{2} = 1$ is a one-parameter family of implicit solutions to $\frac{d y}{d x} = \frac{x y}{x^{2} - 1}$ , for c as an arbitrary non-zero constant.

Step 5: To represent the solution curves on a graph.

When $c = 1$

$y = \sqrt{1 - x^{2}}$ (Represented with red colour)

When $c = - 1$

$y = \sqrt{- (1 - x^{2})}$ (Represented with a red-coloured dotted line)

When $c = 2$

$y = \sqrt{\frac{1 - x^{2}}{2}}$ (Represented with blue colour)

When $c = - 2$

$y = \sqrt{\frac{1 - x^{2}}{(- 2)}}$ (Represented with a blue-coloured dotted line)

When $c = 3$

role="math" localid="1663944317705" $y = \sqrt{\frac{1 - x^{2}}{3}}$ (Represented with orange colour)

When $c = - 3$

$y = \sqrt{\frac{1 - x^{2}}{(- 3)}}$ (Represented with orange-coloured dotted line)

Graph representing the solution curves corresponding to $c = \pm 1, \pm 2, \pm 3$ .

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Verify that $x^{2} + {cy}^{2} = 1,$ where c is an arbitrary non-zero constant, is a one-parameter family of implicit solutions to $\frac{dy}{dx} = \frac{xy}{x^{2} - 1}$ and graph several of the solution curves using the same coordinate axes.

Step by Step Solution

Step 1: Important formula.

Step 2: Taking the given function as a function of x in y.

Step 3: Differentiate the function in step 2, with respect to x.

Step 4: Simplification of the differential equation obtained in step 2.

Step 5: To represent the solution curves on a graph.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Verify that x2+cy2=1, where c is an arbitrary non-zero constant, is a one-parameter family of implicit solutions to dydx=xyx2-1 and graph several of the solution curves using the same coordinate axes.

Step by Step Solution

Step 1: Important formula.

Step 2: Taking the given function as a function of x in y.

Step 3: Differentiate the function in step 2, with respect to x.

Step 4: Simplification of the differential equation obtained in step 2.

Step 5: To represent the solution curves on a graph.

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Verify that $x^{2} + {cy}^{2} = 1,$ where c is an arbitrary non-zero constant, is a one-parameter family of implicit solutions to $\frac{dy}{dx} = \frac{xy}{x^{2} - 1}$ and graph several of the solution curves using the same coordinate axes.