The equation 3y e^x/y dx + (y - 3x e^x/y) dy = 0 is written as dx/dy = G(v), where v = x/y. What is G(v)?

Question 27

HardOriginal question

The equation $3y\,e^{x/y}\,dx + (y - 3x\,e^{x/y})\,dy = 0$ is written as $\frac{dx}{dy} = G(v)$ , where $v = \frac{x}{y}$ . What is $G(v)$ ?

v - \frac{1}{3}e^{-v}

Correct answerYour answer

\frac{1}{3}e^{-v} - v

Correct answerYour answer

3v - e^{-v}

Correct answerYour answer

v - \frac{1}{3}e^{v}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

Move the $dy$ term across, changing its sign: $\frac{dx}{dy} = \frac{3x\,e^{x/y} - y}{3y\,e^{x/y}}$ .
Divide top and bottom by $y\,e^{x/y}$ : $\frac{3v - e^{-v}}{3} = v - \frac{1}{3}e^{-v}$ .

Worth noting

$\frac{1}{3}e^{-v} - v$ keeps the sign of the $dy$ term; moving it across the equals sign flips it.

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