Q2.3-8E Question: In Problems 7-16, obta... [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: Q2.3-8E (page 54)

Question: In Problems 7-16, obtain the general solution to the equation.
$\frac{dy}{dx} = \frac{y}{x} + 2 x + 1$

Short Answer

Expert verified

The general solution to the given equation is $y = 2 x^{2} + xln |x| + Cx$ .

Step by step solution

Method for solving linear equations

Write the equation in the standard form $\frac{dy}{dx} + P (x) y = Q (x)$ .
Calculate the integrating factor by $渭 (x)$ the formula .
Multiply the equation in standard form by $渭 (x)$ and, recalling that the left-hand side is just $\frac{d}{dx} [渭 (x) y]$ , obtain

$渭 (x) \frac{dy}{dx} + P 渭 (x) y = 渭 (x) Q (x) \frac{d}{dx} [渭 (x) y] = 渭 (x) Q (x)$

Integrate the last equation and solve for y by dividing by $渭 (x)$ to obtain $y (x) = \frac{1}{蟺 (x)} [{鈭�}_{}^{} 蟺 (x) Q (x) + c]$ . Here C is an arbitrary constant.

Solve the given equation

Given that,

$\frac{dy}{dx} = \frac{y}{x} + 2 x + 1 . . . . . . . . . (1)$

Write the equation (1) into standard form of linear equation.

$\frac{dy}{dx} = \frac{y}{x} + 2 x + 1 \frac{dy}{dx} - \frac{y}{x} = 2 x + 1 . . . . . . . . (2)$

Calculate the integrating factor of $渭 (x)$ .

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

Then,

$\begin{array}{rcl} 渭 (x) & = & e^{{鈭�}_{}^{} P (x) dx} \\ = & e^{{鈭�}_{}^{} - \frac{1}{x} dx} \\ = & e^{- \ln x} \\ = & \frac{1}{x} \end{array}$

Simplification method

Multiply $渭 (x)$ in equation (2)

$\frac{1}{x} \frac{dy}{dx} - \frac{1}{x} \frac{y}{x} = \frac{1}{x} (2 x + 1) \frac{1}{x} \frac{dy}{dx} - \frac{y}{x^{2}} = 2 + \frac{1}{x} \frac{d}{dx} [\frac{1}{x} y] = 2 + \frac{1}{x}$

Integrating both sides.

$\begin{array}{rcl} {鈭�}_{}^{} \frac{d}{dx} [\frac{1}{x} y] & = & {鈭�}_{}^{} (2 + \frac{1}{x}) dx \\ \frac{y}{x} & = & 2 x + \ln |x| + C \end{array}$

So, the solution is $\begin{array}{rcl} \frac{y}{x} & = & 2 x + \ln | x | + C \end{array}$ .

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91影视

Question: In Problems 7-16, obtain the general solution to the equation.
$\frac{dy}{dx} = \frac{y}{x} + 2 x + 1$

Short Answer

Step by step solution

Method for solving linear equations

Solve the given equation

Simplification method

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

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91影视

Question: In Problems 7-16, obtain the general solution to the equation.dydx=yx+2x+1

Short Answer

Step by step solution

Method for solving linear equations

Solve the given equation

Simplification method

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Probability and Statistics

Decision Maths

Discrete Mathematics

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Question: In Problems 7-16, obtain the general solution to the equation.
$\frac{dy}{dx} = \frac{y}{x} + 2 x + 1$