Chapter 2: Q 9RP (page 79)

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

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, $(x^{2} + y^{2}) dx + 3 xy dy = 0 \cdot \cdot \cdot \cdot \cdot \cdot (1)$

Evaluate the equation (1).

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

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 $dp = 8 t dt$ .

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

Now integrate on both sides.

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

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 ${(x^{2} + 4 y^{2})}^{3} x^{2} = C$

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

Short Answer

Step by step solution

Given information and simplification

Evaluation method

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Question: In Problems 1-30, solve the equation. x2+y2dx+3xy 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

Logic and Functions

Discrete Mathematics

Pure Maths

Probability and Statistics

Mechanics Maths

Decision Maths

Study anywhere. Anytime. Across all devices.

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