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

$x \equiv 0$ In problems 33-40, Solve the equation given in Problem 2.
${(y - 4 x - 1)}^{2} dx - dy = 0$

$\frac{1}{4} \log |\frac{y - 4 x - 3}{y - 4 x + 1}| - x = C$ and

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Solving the given equation

The given equation is,

${(y - 4 x - 1)}^{2} dx - dy = 0$

Simplify the above equation,

$\begin{array}{l} {(y - 4 x - 1)}^{2} dx = dy \\ \frac{dy}{dx} = {(y - 4 x - 1)}^{2} \cdot \cdot \cdot \cdot \cdot \cdot (1) \end{array}$

Let,

$\begin{array}{l} t = y - 4 x - 1 \\ \frac{dt}{dx} = \frac{dy}{dx} - 4 \\ \frac{dy}{dx} = \frac{dt}{dx} + 4 \end{array}$

Substitute the $t = y - 4 x - 1$ and $\frac{dy}{dx} = \frac{dt}{dx} + 4$ in the equation (1),

$\frac{dt}{dx} + 4 = {(t)}^{2}$

Simplify the above equation,

$\frac{dt}{dx} = t^{2} - 4$

Cross multiplication on both sides in the above equation,

$\frac{dt}{t^{2} - 4} = dx$

Integrating

Taking integration of both sides in the above equation

$\int \frac{dt}{t^{2} - 2^{2}} = \int dx$

One knows that,

$\int \frac{dx}{x^{2} - a^{2}} = \frac{1}{2 a} \log |\frac{x - a}{x + a}| + c$

Solve the above equation,

$\begin{array}{l} \frac{1}{2 (2)} \log |\frac{t - 2}{t + 2}| = x + c \\ \frac{1}{4} \log |\frac{t - 2}{t + 2}| - x = c \end{array}$

Finding the solution

Substitute the $t = y - 4 x - 1$ in the above equation,

$\frac{1}{4} \log |\frac{(y - 4 x - 1) - 2}{(y - 4 x - 1) + 2}| - x = c$

Simplify the above equation,

$\frac{1}{4} \log |\frac{y - 4 x - 3}{y - 4 x + 1}| - x = C$

Also, note that $x \equiv 0$ is a solution.

Hence the solution is $\frac{1}{4} \log |\frac{y - 4 x - 3}{y - 4 x + 1}| - x = C$ and $x \equiv 0$

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

$x \equiv 0$ In problems 33-40, Solve the equation given in Problem 2.
${(y - 4 x - 1)}^{2} dx - dy = 0$

Step by Step Solution

Solving the given equation

Integrating

Finding the solution

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$x \equiv 0$ In problems 33-40, Solve the equation given in Problem 2.
${(y - 4 x - 1)}^{2} dx - dy = 0$