Q 8RP Question: In Problems 聽, solve ... [FREE SOLUTION]

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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 2: Q 8RP (page 79)

Question: In Problems , solve the equation.
$\frac{dy}{dx} + \frac{2 y}{x} = 2 x^{2} y^{2}$

Short Answer

Expert verified

The solution of the given equation is $y = {(- 2 x^{3} + {Cx}^{2})}^{- 1}$ .

Step by step solution

Given information and simplification

Given that, $\frac{dy}{dx} + \frac{2 y}{x} = 2 x^{2} y^{2} 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (1)$

Multiply by y^-2 in equation (1),

$y^{- 2} \frac{dy}{dx} + \frac{2}{xy} = 2 x^{2} 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (2)$

Now substitute u = y^-1 differentiate with respect to x.

$\begin{matrix} \frac{du}{dx} = - y^{- 2} \frac{dy}{dx} \\ - \frac{du}{dx} = y^{- 2} \frac{dy}{dx} \end{matrix}$

Substitute in equation (2).

$\begin{matrix} - \frac{du}{dx} + \frac{2}{x} u = 2 x^{2} \\ \frac{du}{dx} - \frac{2}{x} u = - 2 x^{2} 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (3) \end{matrix}$

Let $P (x) = - \frac{2}{x}$

Find the value of $渭 (x)$ .

$\begin{matrix} 渭 (x) = e^{鈭� P (x) dx} \\ = e^{- 鈭� \frac{2}{x} dx} \\ = e^{- 2 \ln |x|} \\ = x^{- 2} \end{matrix}$

Evaluation method

Multiply x^-2 in equation (3) on both sides.

$\begin{matrix} x^{- 2} \frac{du}{dx} - x^{- 2} \frac{2}{x} u = - 2 \\ \frac{1}{x^{2}} \frac{du}{dx} - \frac{2}{x^{3}} u = - 2 \\ \frac{d}{dx} [\frac{1}{x^{2}} u] = - 2 \end{matrix}$

Now integrate the equation on both sides.

$\begin{matrix} 鈭� \frac{d}{dx} [\frac{1}{x^{2}} u] dx = 鈭� - 2 鈥� dx \\ \frac{1}{x^{2}} u = - 2 x + C \\ u = - 2 x^{3} + {Cx}^{2} \end{matrix}$

Substitute $u = y^{- 1}$ , we get,

$\begin{matrix} y^{- 1} = - 2 x^{3} + {Cx}^{2} \\ y = {(- 2 x^{3} + {Cx}^{2})}^{- 1} \end{matrix}$

So, the solution is $y = {(- 2 x^{3} + {Cx}^{2})}^{- 1}$

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91影视

Question: In Problems , solve the equation.
$\frac{dy}{dx} + \frac{2 y}{x} = 2 x^{2} y^{2}$

Short Answer

Step by step solution

Given information and simplification

Evaluation method

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91影视

Question: In Problems , solve the equation.dydx+2yx=2x2y2

Short Answer

Step by step solution

Given information and simplification

Evaluation method

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

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Applied Mathematics

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Theoretical and Mathematical Physics

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Question: In Problems , solve the equation.
$\frac{dy}{dx} + \frac{2 y}{x} = 2 x^{2} y^{2}$