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

Use the method discussed under “Homogeneous Equations” to solve problems 9-16 $(xy + y^{2}) dx - x^{2} dy = 0$

Homogeneous equation for the given equation is $y = \frac{- x}{\ln |x| + C}$ and $y = 0$ .

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Step by Step Solution

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General form of Homogeneous equation

If the right-hand side of the equation $\frac{dy}{dx} = f (x, y)$ can be expressed as a function of the ratio $\frac{y}{x}$ alone, then we say the equation is homogeneous.

Evaluate the given equation

Given, $(xy + y^{2}) dx - x^{2} dy = 0$

By Evaluating,

$\begin{array}{rcl} (xy + y^{2}) dx - x^{2} dy & = & 0 \\ - x^{2} dy & = & - (xy + y^{2}) dx \\ \frac{dy}{dx} & = & \frac{xy + y^{2}}{x^{2}} \\ = & \frac{y}{x} + \frac{y^{2}}{x^{2}} \\ = & \frac{y}{x} + {(\frac{y}{x})}^{2} \end{array}$

Substitution method

Let us take $v = \frac{y}{x}$ $v = \frac{y}{x}$

Then $y = v x$

By Differentiating,

$\frac{dy}{dx} = v + x \frac{dv}{dx} v + v^{2} = v + x \frac{dv}{dx} v^{2} = x \frac{dv}{dx} \frac{1}{v^{2}} dv = \frac{1}{x} dx$

Now integrating on both sides,

$\begin{array}{rcl} \int v^{- 2} dv & = & \int \frac{1}{x} dx \\ - v^{- 1} & = & \ln |x| + C \\ - \frac{1}{v} & = & \ln |x| + C \end{array}$

Substitute $v = \frac{y}{x}$

$- \frac{1}{\frac{y}{x}} = I n |x| + C - \frac{x}{y} = I n |x| + C y = \frac{x}{I n |x| + C}, y = 0$

Therefore, Homogeneous equation for the given equation is $y = \frac{- x}{\ln |x| + C}$ and localid="1655121736301" $y = 0$ .

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

Use the method discussed under “Homogeneous Equations” to solve problems 9-16 $(xy + y^{2}) dx - x^{2} dy = 0$

Step by Step Solution

General form of Homogeneous equation

Evaluate the given equation

Substitution method

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Use the method discussed under “Homogeneous Equations” to solve problems 9-16 xy+y2 dx-x2dy=0

Step by Step Solution

General form of Homogeneous equation

Evaluate the given equation

Substitution method

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Use the method discussed under “Homogeneous Equations” to solve problems 9-16 $(xy + y^{2}) dx - x^{2} dy = 0$