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

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

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given information and simplification

Given that, $\frac{dy}{dx} - \frac{2 y}{x} = x^{2} \cos x, y (π) = 2 \cdot \cdot \cdot \cdot \cdot \cdot (1)$

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

Find the value of $μ (x)$ .

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

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

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

Step 2: integration method

Now integrate the equation on both sides.

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

So, the solution is found.

Step 3: Find the initial value

Given that, $y (π) = 2$ .

Then, $x = π$ and y = 2.

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

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

Substitute the value of C in equation (2).

$y = x^{2} \sin x + \frac{2 x^{2}}{π^{2}}$

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

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

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

Step by Step Solution

Step 1: Given information and simplification

Step 2: integration method

Step 3: Find the initial value

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

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Step by Step Solution

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