The equation (x^2 - 2y^2) dx + xy dy = 0 is written as dy/dx = F(v), where v = y/x. What is F(v)?

Homogeneous & Linear Differential Equations · Question 24 of 120

Question 24

MediumOriginal question

The equation $(x^{2} - 2y^{2})\,dx + xy\,dy = 0$ is written as $\frac{dy}{dx} = F(v)$ , where $v = \frac{y}{x}$ . What is $F(v)$ ?

\frac{1}{v} - 2v

Correct answerYour answer

2v^{2} - 1

Correct answerYour answer

2v - \frac{1}{v}

Correct answerYour answer

2v + \frac{1}{v}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

$xy\,dy = -(x^{2} - 2y^{2})\,dx$ , so $\frac{dy}{dx} = \frac{2y^{2} - x^{2}}{xy}$ .
Divide top and bottom by $x^{2}$ : $\frac{2v^{2} - 1}{v} = 2v - \frac{1}{v}$ .

Worth noting

$\frac{1}{v} - 2v$ forgets the minus sign that comes from moving the $dx$ term across.

Ten questions from Homogeneous & Linear Differential Equations, at your own pace.

Ten questions from Homogeneous & Linear Differential Equations, at your own pace.