Q 19RP 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 19RP (page 79)

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

Short Answer

Expert verified

The solution of the given equation is $y^{2} = {Cx}^{3} + x^{2}$ and x = 0.

Step by step solution

Given information and simplification

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

Evaluate the equation (1).

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

$\frac{dy}{dx} = - \frac{x^{2} - 3 y^{2}}{2 xy} 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (2)$

Since the given equation is homogeneous. Then, put y = tx and $\frac{dy}{dx} = x \frac{dt}{dx} + t$ .

Substitute the values in equation (2).

role="math" localid="1663948749390" $\begin{matrix} \frac{dy}{dx} = - \frac{x^{2} - 3 y^{2}}{2 xy} \\ x \frac{dt}{dx} + t = - \frac{(1 - 3 t^{2})}{2 t} \\ x \frac{dt}{dx} = - \frac{(1 - 3 t^{2})}{2 t} - t \end{matrix} \begin{matrix} x \frac{dt}{dx} = \frac{- 1 + 3 t^{2} - 2 t^{2}}{2 t} \\ = \frac{- 1 + t^{2}}{2 t} \\ \frac{2 t}{t^{2} - 1} dt = \frac{dx}{x} \end{matrix}$

Let us assume p = t² - 1. Differentiate with respect to x.

dp = 2t dt .

Substitute the values in equation (1)

$\frac{1}{p} dp = \frac{dx}{x}$

Evaluation method

Now integrate the equation (2) on both sides.

$\begin{matrix} 鈭� \frac{1}{p} dp = 鈭� \frac{dx}{x} \\ \ln |p| = \ln |x| + C \\ p = Cx \end{matrix}$

Substitute the value of p.

$t^{2} - 1 = Cx$

Substitute the value of t.

$\begin{matrix} y^{2} - x^{2} = {Cx}^{3} \\ y^{2} = {Cx}^{3} + x^{2} \end{matrix}$

So, the solution is $y^{2} = {Cx}^{3} + x^{2}$

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91影视

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

Short Answer

Step by step solution

Given information and simplification

Evaluation method

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91影视

Question: In Problems 1-30, solve the equation.x2-3y2dx+2xy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

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Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

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