Chapter 2: Q 6RP (page 79)

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

Short Answer

Expert verified

The solution of the given equation is $y^{- 2} = 2 \ln |1 - x^{2}| + C$ .

Step by step solution

Given information and simplification

Given that, $2 {xy}^{3} dx - (1 - x^{2}) dy = 0 \cdot \cdot \cdot \cdot \cdot \cdot (1)$

Evaluate the equation (1).

$\begin{matrix} 2 {xy}^{3} dx - (1 - x^{2}) dy = 0 \\ - (1 - x^{2}) dy = - 2 {xy}^{3} dx \\ \frac{1}{y^{3}} dy = \frac{2 x}{1 - x^{2}} dx \end{matrix}$

$\frac{1}{y^{3}} dy = \frac{2 x}{1 - x^{2}} dx \cdot \cdot \cdot \cdot \cdot \cdot (2)$

Now integrate the equation (2) on both sides.

$\int \frac{1}{y^{3}} dy = \int \frac{2 x}{1 - x^{2}} dx \cdot \cdot \cdot \cdot \cdot \cdot (3)$

Evaluation method

Find the value of $\int \frac{2 x}{1 - x^{2}} dx$ separately.

Let us take $u = 1 - x^{2}, dx = - \frac{1}{2 x} du$ .

Substitute the values.

$\begin{matrix} \int \frac{2 x}{1 - x^{2}} dx = \int \frac{2 x}{u} \frac{- 1}{2 x} du \\ = - \int \frac{1}{u} du \\ = - \ln |u| + C_{1} \\ = - \ln |1 - x^{2}| + C_{1} \end{matrix}$

Use the value in equation (3).

$\begin{matrix} \int \frac{1}{y^{3}} dy = \int \frac{2 x}{1 - x^{2}} dx \\ - \frac{y^{- 2}}{2} = - \ln |1 - x^{2}| + C_{1} \\ y^{- 2} = 2 \ln |1 - x^{2}| + C \end{matrix}$

So, the solution is $y^{- 2} = 2 \ln |1 - x^{2}| + C$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

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

Pure Maths

Geometry

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Question: In Problems 1-30, solve the equation.2xy3dx-1-x2dy=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

Pure Maths

Geometry

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

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