Q 9RP Question: In Problems 1-30, solv... [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 9RP (page 79)

Question: In Problems 1-30, solve the equation.
$鈥� (x^{2} + y^{2}) dx + 3 xy 鈥� dy = 0$

Short Answer

Expert verified

The solution of the given equation is ${(x^{2} + 4 y^{2})}^{3} x^{2} = C$

Step by step solution

Given information and simplification

Given that, $(x^{2} + y^{2}) dx + 3 xy 鈥� dy = 0 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (1)$

Evaluate the equation (1).

$\begin{matrix} (x^{2} + y^{2}) dx + 3 xy 鈥� dy = 0 \\ 3 xy 鈥� dy = - (x^{2} + y^{2}) dx \\ \frac{dy}{dx} = - \frac{x^{2} + y^{2}}{3 xy} \end{matrix}$

Let us take $y = tx$ . Differentiate with respect to x.

$\frac{dy}{dx} = x \frac{dt}{dx} + t$

Then,

$\begin{matrix} x \frac{dt}{dx} + t = - \frac{(1 + t^{2})}{3 t} \\ x \frac{dt}{dx} = - \frac{(1 + t^{2})}{3 t} - t \\ x \frac{dt}{dx} = \frac{- (1 + t^{2}) - 3 t^{2}}{3 t} \end{matrix}$

$\begin{matrix} x \frac{dt}{dx} = \frac{- 1 - 4 t^{2}}{3 t} \\ \frac{3 t}{1 + 4 t^{2}} dt = - \frac{dx}{x} \end{matrix}$

Let us take $p = 1 + 4 t^{2}$ and $dp = 8 t 鈥� dt$ .

$\frac{3}{8 p} dp = - \frac{dx}{x}$

Now integrate on both sides.

$\begin{matrix} 鈭� \frac{3}{8 p} dp = - 鈭� \frac{dx}{x} \\ \frac{3}{8} \ln |p| = - \ln |x| + C \\ p = C' x^{- \frac{8}{3}} \end{matrix}$

Evaluation method

Substitute the value of p.

$\begin{matrix} 1 + 4 t^{2} = C' x^{- \frac{8}{3}} \\ {(1 + 4 t^{2})}^{3} = C'' x^{- 8} \end{matrix}$

Substitute the value of t.

$\begin{matrix} {(\frac{x^{2} + 4 y^{2}}{x^{2}})}^{3} = C'' x^{- 8} \\ {(x^{2} + 4 y^{2})}^{3} x^{2} = C \end{matrix}$

So, the solution is ${(x^{2} + 4 y^{2})}^{3} x^{2} = C$

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

Question: In Problems 1-30, solve the equation.
$鈥� (x^{2} + y^{2}) dx + 3 xy 鈥� dy = 0$

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

Mechanics Maths

Logic and Functions

Statistics

Discrete Mathematics

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

91影视

Question: In Problems 1-30, solve the equation.鈥�x2+y2dx+3xy鈥�dy=0

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

Mechanics Maths

Logic and Functions

Statistics

Discrete Mathematics

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Question: In Problems 1-30, solve the equation.
$鈥� (x^{2} + y^{2}) dx + 3 xy 鈥� dy = 0$