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 problem , solve the equation. $(x + x y^{2}) d x + e^{x^{2}} y d y = 0$

The solution of the given differential equation is . $y = \pm \sqrt{C e x p [e x p (- x^{2})] - 1}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Concept of Separable Differential Equation

A first-order ordinary differential equation $\frac{d y}{d x} = f (x, y)$ is referred to as separable if the function in the right-hand side of the equation is expressed as a product of two functions g(x) that is a function of x alone and h(y) that is a function of y alone.

A separable differential equation can be expressed as $\frac{d y}{d x} = g (x) h (y)$ . By separating the variables, the equation follows $\frac{d y}{h (y)} = g (x) d x$ . Then, on direct integration of both sides, the solution of the differential equation is determined.

Step 2: Solution of the Equation

The given equation is

$(x + x y^{2}) d x + e^{x^{2}} y d y = 0 . . . . . (1)$

Re-write equation (1) as follows:

$(x + x y^{2}) d x + e^{x^{2}} y d y = 0 \frac{d y}{d x} = - \frac{x (1 + y^{2})}{e^{x^{2}} y} . . . . . . . . (2)$

After separating the variables, equation (2) can be written as

$\frac{y}{1 + y^{2}} d y = - x e^{x^{2}} d x . . . . . . . (3)$

Integrate both sides of equation (3). It results,

$\int \frac{y}{1 + y^{2}} d y = \int - x e^{x^{2}} d x \frac{1}{2} \int \frac{2 y}{1 + y^{2}} d y = \frac{1}{2} \int - 2 x e^{x^{2}} d x \frac{1}{2} \int \frac{d (1 + y^{2})}{1 + y^{2}} = \frac{1}{2} \int 2 e^{x^{2}} d (- x^{2}) \frac{1}{2} \ln (1 + y^{2}) = \frac{1}{2} e^{- x^{2}} + \ln k [\ln k = I n t e g r a t i n g c o n s \tan t] \ln (1 + y^{2}) = e^{- x^{2}} + \ln C [\ln C = 2 \ln k = c o n s \tan t] \ln (\frac{1 + y^{2}}{C}) = e^{x^{2}} 1 + y^{2} = C e^{x^{2}} y = \pm \sqrt{C e^{x^{2}} - 1} y = \pm \sqrt{C e x p [e x p (x^{2})] - 1}$

Therefore, the solution of the given equation is $y = \pm \sqrt{C e x p [e x p (x^{2})] - 1}$ .

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

In problem , solve the equation. $(x + x y^{2}) d x + e^{x^{2}} y d y = 0$

Step by Step Solution

Step 1: Concept of Separable Differential Equation

Step 2: Solution of the Equation

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

In problem , solve the equation. (x+xy2)dx+ex2y dy=0

Step by Step Solution

Step 1: Concept of Separable Differential Equation

Step 2: Solution of the Equation

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In problem , solve the equation. $(x + x y^{2}) d x + e^{x^{2}} y d y = 0$