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: Q9E (page 76)

Use the method discussed under 鈥淗omogeneous Equations鈥� to solve problems 9 - 16.
$(xy + y^{2}) dx - x^{2} dy = 0$

Short Answer

Expert verified

Homogeneous equation for the given equation is $y = \frac{- x}{\ln | x | + C}$ and $y 鈮� 0$ .

Step by step solution

General form of Homogeneous equation

If the right-hand side of the equation $\frac{dy}{dx} = f (x, y)$ can be expressed as a function of the ratio $\frac{y}{x}$ alone, then we say the equation is homogeneous.

Evaluate the given equation

Given, $(xy + y^{2}) dx - x^{2} dy = 0$ .

By Evaluating,

role="math" localid="1654867974528" $\begin{array}{rcl} (xy + y^{2}) dx - x^{2} dy & = & 0 \\ - x^{2} dy & = & - (xy + y^{2}) dx \\ \frac{dy}{dx} & = & \frac{(xy + y^{2})}{x^{2}} \\ = & \frac{y}{x} + \frac{y^{2}}{x^{2}} \\ = & \frac{y}{x} + {(\frac{y}{x})}^{2} \end{array}$

Substitution method

Let us take $v = \frac{y}{x}$ .

Then $y = vx$ .

By Differentiating,

$\begin{array}{rcl} \frac{dy}{dx} & = & v + x \frac{dv}{dx} \\ v + v^{2} & = & v + x \frac{dv}{dx} \\ v^{2} & = & x \frac{dv}{dx} \\ \frac{1}{v^{2}} dv & = & \frac{1}{x} dx \end{array}$

Now integrating on both sides,

role="math" localid="1654868488187" $\begin{array}{rcl} 鈭� v^{- 2} dv & = & 鈭� \frac{1}{x} dx \\ - v^{- 1} & = & \ln |x| + C \\ - \frac{1}{v} & = & \ln |x| + C \end{array}$

Substitute $v = \frac{y}{x}$ .

$\begin{array}{rcl} - \frac{1}{\frac{y}{x}} & = & \ln |x| + C \\ - \frac{x}{y} & = & \ln |x| + C \\ y & = & - \frac{x}{\ln |x| + C}, y 鈮� 0 \end{array}$

Therefore, Homogeneous equation for the given equation is $y = \frac{- x}{\ln | x | + C}$ and $y 鈮� 0$ .

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

Use the method discussed under 鈥淗omogeneous Equations鈥� to solve problems 9 - 16.
$(xy + y^{2}) dx - x^{2} dy = 0$

Short Answer

Step by step solution

General form of Homogeneous equation

Evaluate the given equation

Substitution method

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

91影视

Use the method discussed under 鈥淗omogeneous Equations鈥� to solve problems 9 - 16. (xy+y2)dx-x2dy=0

Short Answer

Step by step solution

General form of Homogeneous equation

Evaluate the given equation

Substitution method

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Use the method discussed under 鈥淗omogeneous Equations鈥� to solve problems 9 - 16.
$(xy + y^{2}) dx - x^{2} dy = 0$