Q 34RP Question: In Problems 31-40, sol... [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 34RP (page 79)

Question: In Problems 31-40, solve the initial value problem.
$\frac{dy}{dx} - \frac{2 y}{x} = x^{2} cosx, 鈥夆赌夆赌夆赌� y (蟺) = 2$

Short Answer

Expert verified

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

Step by step solution

Given information and simplification

Given that, $\frac{dy}{dx} - \frac{2 y}{x} = x^{2} \cos 鈥� x, 鈥夆赌夆赌夆赌� y (蟺) = 2 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (1)$

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

Find the value of $渭 (x)$ .

$\begin{matrix} 渭 (x) = e^{鈭� P (x) dx} \\ = e^{- 鈭� \frac{2}{x} dx} \\ = e^{- 2 \ln |x|} \\ = \frac{1}{x^{2}} \end{matrix}$

Multiply $\frac{1}{x^{2}}$ in equation (1) on both sides.

$\begin{matrix} \frac{1}{x^{2}} \frac{dy}{dx} - \frac{2 y}{x^{3}} = \cos 鈥� x \\ \frac{dy}{dx} [\frac{1}{x^{2}} y] = \cos 鈥� x \end{matrix}$

integration method

Now integrate the equation on both sides.

${\begin{matrix} 鈭� \frac{dy}{dx} [\frac{1}{x^{2}} y] dx = 鈭� \cos 鈥� x 鈥� dx \\ \frac{1}{x^{2}} y = \sin 鈥� x + C_{1} \\ y = x^{2} \sin 鈥� x + x^{2} C 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (2) \end{matrix}}_{}$

So, the solution is found.

Find the initial value

Given that, $y (蟺) = 2$ .

Then, $x = 蟺$ and y = 2.

Substitute the value in equation (2) to get the value of C.

$\begin{matrix} y = x^{2} \sin 鈥� x + x^{2} C \\ 2 = 蟺^{2} \sin (蟺) + 蟺^{2} C \\ \frac{2}{蟺^{2}} = C \end{matrix}$

Substitute the value of C in equation (2).

$y = x^{2} \sin 鈥� x + \frac{2 x^{2}}{蟺^{2}}$

So, the solution is $y = x^{2} \sin 鈥� x + \frac{2 x^{2}}{蟺^{2}}$

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91影视

Question: In Problems 31-40, solve the initial value problem.
$\frac{dy}{dx} - \frac{2 y}{x} = x^{2} cosx, 鈥夆赌夆赌夆赌� y (蟺) = 2$

Short Answer

Step by step solution

Given information and simplification

integration method

Find the initial value

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Most popular questions from this chapter

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91影视

Question: In Problems 31-40, solve the initial value problem.dydx-2yx=x2cosx,鈥夆赌夆赌夆赌�y蟺=2

Short Answer

Step by step solution

Given information and simplification

integration method

Find the initial value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Probability and Statistics

Geometry

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.

Question: In Problems 31-40, solve the initial value problem.
$\frac{dy}{dx} - \frac{2 y}{x} = x^{2} cosx, 鈥夆赌夆赌夆赌� y (蟺) = 2$