Q3E In Problems , identify the equat... [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: Q3E (page 69)

In Problems , identify the equation as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.
$鈥� (2 x + y) dx + (x - 2 y) dy = 0$

Short Answer

Expert verified

The given equation is exact.

Step by step solution

General form of separable, linear, exact or integrating factors

Separable equation:If the right-hand side of the equation $\frac{dy}{dx} = f (x, y)$ can be expressed as a function g(x) that depends only on x times a function p(y) that depends only on y, then the differential equation is called separable.

Linear equation:Standard form of linear equation is $\frac{dy}{dx} + P (x) y = Q (x)$ .

Exact differential form:The differential form $M (x, y) dx + N (x, y) dy$ is said to be exact in a rectangle R if there is a function such that

$\frac{鈭� F}{鈭� x} (x, y) = M (x, y) 鈥� 鈥� 鈥� 鈥� 鈥� and 鈥� 鈥� 鈥� 鈥� \frac{鈭� F}{鈭� y} (x, y) = N (x, y)$

Special integrating factors: If $\frac{(\frac{鈭� M}{鈭� y} - \frac{鈭� N}{鈭� x})}{N}$ is continuous and depends only on x. If $\frac{(\frac{鈭� N}{鈭� x} - \frac{鈭� M}{鈭� y})}{M}$ is continuous and depends only on y.

Evaluate the given equation

Given, $(2 x + y) dx + (x - 2 y) dy = 0$ .

Evaluate it.

$\begin{matrix} (2 x + y) dx + (x - 2 y) dy = 0 \\ \frac{dy}{dx} = \frac{2 x + y}{x - 2 y} \end{matrix}$

Compare the equation with general form of separable and linear equation.

So, the given equation is neither separable nor linear.

Testing for exactness

Given, $(2 x + y) dx + (x - 2 y) dy = 0$ .

Let $M = 2 x + y, N = x - 2 y$ .

Then,

$\begin{matrix} \frac{鈭� M}{鈭� y} = 1 \\ \frac{鈭� N}{鈭� x} = 1 \end{matrix}$

So, $\frac{鈭� M}{鈭� y} = \frac{鈭� N}{鈭� x}$ .

Therefore, the given equation is exact.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

In Problems , identify the equation as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.
$鈥� (2 x + y) dx + (x - 2 y) dy = 0$

Short Answer

Step by step solution

General form of separable, linear, exact or integrating factors

Evaluate the given equation

Testing for exactness

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Mechanics Maths

Probability and Statistics

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

In Problems , identify the equation as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.鈥�2x+ydx+x-2ydy=0

Short Answer

Step by step solution

General form of separable, linear, exact or integrating factors

Evaluate the given equation

Testing for exactness

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Mechanics Maths

Probability and Statistics

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

In Problems , identify the equation as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.
$鈥� (2 x + y) dx + (x - 2 y) dy = 0$