Q 13RP Question: In Problems 聽1 - 30, ... [FREE SOLUTION]

91影视

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 13RP (page 79)

Question: In Problems 1 - 30, solve the equation.
$\frac{dy}{dx} - \frac{y}{x} = x^{2} \sin 鈥� 2 x$

Short Answer

Expert verified

The solution of the given equation is $y = - \frac{x^{2} \cos 鈥� 2 x}{2} + \frac{x 鈥� \sin 鈥� 2 x}{4} + Cx$ .

Step by step solution

Given information and simplification

Given that,

$\frac{dy}{dx} - \frac{y}{x} = x^{2} 鈥� \sin 鈥� 2 x 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (1)$

Let $P (x) = - \frac{1}{x}$ .

Find the value of $渭 (x)$ .

$\begin{matrix} 渭 (x) = e^{鈭� P (x) dx} \\ = e^{- 鈭� \frac{1}{x} dx} \\ = e^{- \ln |x|} \\ = \frac{1}{x} \end{matrix}$

Multiply $\frac{1}{x}$ in equation (1) on both sides.

$\begin{matrix} \frac{1}{x} \frac{dy}{dx} - \frac{y}{x^{2}} = x 鈥� \sin 鈥� 2 x \\ \frac{d}{dx} [\frac{1}{x} y] = x 鈥� \sin 鈥� 2 x \end{matrix}$

Now integrate the equation on both sides.

$鈭� \frac{d}{dx} [\frac{1}{x} y] dx = 鈭� x 鈥� \sin 鈥� 2 x 鈥� dx 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (2)$

Evaluation method

Find the value of $鈭� x 鈥� \sin 鈥� 2 x 鈥� dx$ separately.

Let us take $u = x, dv = \sin 鈥� 2 x 鈥� dx$ .

$du = dx, v = - \frac{\cos 鈥� 2 x}{2}$

Use the integration by parts formula.

$\begin{matrix} 鈭� x 鈥� \sin 鈥� 2 x 鈥� dx = - \frac{x 鈥� \cos 鈥� 2 x}{2} + \frac{1}{2} 鈭� \cos 鈥� 2 x 鈥� dx \\ = - \frac{x 鈥� \cos 鈥� 2 x}{2} + \frac{\sin 鈥� 2 x}{4} + C_{1} \end{matrix}$

Now substitute in equation (2)

$\begin{matrix} 鈭� \frac{d}{dx} [\frac{1}{x} y] dx = 鈭� x 鈥� \sin 鈥� 2 x 鈥� dx \\ \frac{1}{x} y = - \frac{x 鈥� \cos 鈥� 2 x}{2} + \frac{\sin 鈥� 2 x}{4} + C_{1} \\ y = - \frac{x^{2} \cos 鈥� 2 x}{2} + \frac{x 鈥� \sin 鈥� 2 x}{4} + Cx \end{matrix}$

So, the solution is $y = - \frac{x^{2} \cos 鈥� 2 x}{2} + \frac{x 鈥� \sin 鈥� 2 x}{4} + Cx$

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91影视

Question: In Problems 1 - 30, solve the equation.
$\frac{dy}{dx} - \frac{y}{x} = x^{2} \sin 鈥� 2 x$

Short Answer

Step by step solution

Given information and simplification

Evaluation method

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91影视

Question: In Problems 1 - 30, solve the equation.dydx-yx=x2sin鈥�2x

Short Answer

Step by step solution

Given information and simplification

Evaluation method

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Discrete Mathematics

Calculus

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Question: In Problems 1 - 30, solve the equation.
$\frac{dy}{dx} - \frac{y}{x} = x^{2} \sin 鈥� 2 x$