Put y = vx in dy/dx = (x^2 + 2y^2)/xy. What does xdv/dx equal?

Homogeneous & Linear Differential Equations · Question 33 of 120

Question 33

EasyOriginal question

Put $y = vx$ in $\frac{dy}{dx} = \frac{x^{2} + 2y^{2}}{xy}$ . What does $x\frac{dv}{dx}$ equal?

\frac{1 + 2v^{2}}{v}

Correct answerYour answer

\frac{1 + v^{2}}{v}

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\frac{1 + 3v^{2}}{v}

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\frac{1 - v^{2}}{v}

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Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

The right side is $\frac{1 + 2v^{2}}{v}$ , and $\frac{dy}{dx} = v + x\frac{dv}{dx}$ .
$x\frac{dv}{dx} = \frac{1 + 2v^{2}}{v} - v = \frac{1 + v^{2}}{v}$ .

Worth noting

$\frac{1 + 2v^{2}}{v}$ is the whole of $\frac{dy}{dx}$ ; the $v$ from $v + x\frac{dv}{dx}$ must be taken away.

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

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