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 $y^{'} = G (ax + by)$ . $\frac{dy}{dx} + \frac{y}{x} = x^{3} y^{2}$

The given equation is the form of Bernoulli equation.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

General form of homogeneous, Bernoulli, linear coefficients 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,

$\frac{d y}{d x} + \frac{y}{x} = x^{3} y^{2}$

By evaluating,

role="math" localid="1655102457300" $\begin{array}{rcl} \frac{d y}{d x} + \frac{y}{x} & = & x^{3} y^{2} \\ \frac{d y}{d x} + (\frac{1}{x}) y & = & (x^{3}) y^{2} \end{array}$

Comparing with general form of Bernoulli equation it seems that the given equation also Bernoulli equation.

Therefore, the given equation is the form of Bernoulli equation.

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Fundamentals Of Differential Equations And Boundary Value Problems

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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 $y^{'} = G (ax + by)$ . $\frac{dy}{dx} + \frac{y}{x} = x^{3} y^{2}$

Step by Step Solution

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

Evaluate the given equation

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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). dydx+yx=x3y2

Step by Step Solution

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

Evaluate the given equation

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In problems, 1-8 identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form $y^{'} = G (ax + by)$ . $\frac{dy}{dx} + \frac{y}{x} = x^{3} y^{2}$