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

In problems 1 - 8 identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form $Y^{'} = G (ax + by)$ . ${(y - 4 x - 1)}^{2} dx - dy = 0$ .

The given equation is the form of linear coefficient.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

General form of homogeneous, Bernoulli, linear coefficients or of the form of Y'=Gax+by

Homogeneous equation

If the right-hand side of the equation

\frac{d y}{d x} = f (x, y)

can be expressed as a function of the ratio

\frac{y}{x}

alone, then we say the equation is homogeneous.

Equations of the form $\frac{d y}{d x} = G (a x + b y)$

When the right-hand side of the equation $\frac{d y}{d x} = f (x, y)$ can be expressed as a function of the combination $a x + b y$ , where a and b are constants, that is, $\frac{d y}{d x} = G (a x + b y)$ then the substitution $z = a x + b y$ transforms the equation into a separable one.

Bernoulli’s equation

A first-order equation that can be written in the form $\frac{d y}{d x} + P (x) y = Q (x) y^{n}$ , where P(x) and Q(x) are continuous on an interval (a,b) and n is a real number, is called a Bernoulli equation.

Equation of Linear coefficients

We have used various substitutions for y to transform the original equation into a new equation that we could solve. In some cases, we must transform both x and y into new variables, say u and v. This is the situation for equations with linear coefficients-that is, equations of the form

$(a_{1} x + b_{1} y + c_{1}) d x + (a_{2} x + b_{2} y + c_{2}) d y = 0$

Evaluate the given equation

Given, ${(y - 4 x - 1)}^{2} d x - d y = 0$

By Evaluating,

role="math" localid="1655099619700" $\begin{array}{rcl} {(y - 4 x - 1)}^{2} d x - d y & = & 0 \\ \frac{d y}{d x} & = & {(y - 4 x - 1)}^{2} \end{array}$

Let $u = y - 4 x$ . Then, differentiate it to find the value of $\frac{d u}{d x}$

$\frac{d u}{d x} = \frac{d y}{d x} - 4 \frac{d y}{d x} = \frac{d u}{d x} + 4$

Now,

$\begin{array}{rcl} \frac{d u}{d x} + 4 & = & {(u - 1)}^{2} \\ \frac{d u}{d x} & = & (u^{2} - 2 u + 1) - 4 \\ = & u^{2} - 2 u - 3 \\ \frac{d u}{u^{2} - 2 u - 3} & = & d x \end{array}$ .....................(1)

Substitution method

Integration equation (1),

$\int \frac{d u}{u^{2} - 2 u - 3} = \int d x I n [{[\frac{3 - u}{u + 1}]}^{\frac{1}{4}}] = x + C$

role="math" localid="1655100433864" $\begin{array}{rcl} \frac{3 - u}{u + 1} & = & C e^{4 x} \\ - 1 + \frac{4}{u + 1} & = & C e^{4 x} \\ \frac{4}{u + 1} & = & C e^{4 x} + 1 \\ u + 1 & = & \frac{4}{C e^{4 x} + 1} \\ u & = & \frac{4}{C e^{4 x} + 1} - 1 \end{array}$

substitute $u = y - 4 x$

role="math" localid="1655100444784" $\begin{array}{rcl} y - 4 x & = & \frac{4}{C e^{4 x} + 1} - 1 \\ y & = & \frac{4}{C e^{4 x} + 1} + 4 x - 1 \end{array}$

It seems that the given equation is linear coefficient.

Therefore, the given equation is the form of linear coefficient.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In problems 1 - 8 identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form $Y^{'} = G (ax + by)$ . ${(y - 4 x - 1)}^{2} dx - dy = 0$ .

Step by Step Solution

General form of homogeneous, Bernoulli, linear coefficients or of the form of Y'=Gax+by

Evaluate the given equation

Substitution method

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In problems 1 - 8 identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form Y'=G(ax+by). (y-4x-1)2dx-dy=0.

Step by Step Solution

General form of homogeneous, Bernoulli, linear coefficients or of the form of Y'=Gax+by

Evaluate the given equation

Substitution method

Most popular questions for Math Textbooks

Want to see more solutions like these?

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In problems 1 - 8 identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form $Y^{'} = G (ax + by)$ . ${(y - 4 x - 1)}^{2} dx - dy = 0$ .