A homogeneous equation dx/dy = G(x/y) is solved with x = vy. Which separated form does it take?

Question 34

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A homogeneous equation $\frac{dx}{dy} = G\left(\frac{x}{y}\right)$ is solved with $x = vy$ . Which separated form does it take?

\frac{dv}{G(v) - v} = \frac{dx}{x}

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\frac{dv}{G(v)} = \frac{dy}{y}

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\frac{dv}{G(v) + v} = \frac{dy}{y}

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\frac{dv}{G(v) - v} = \frac{dy}{y}

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Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$x = vy$ gives $\frac{dx}{dy} = v + y\frac{dv}{dy}$ , so $y\frac{dv}{dy} = G(v) - v$ .
Separate: $\frac{dv}{G(v) - v} = \frac{dy}{y}$ ; $y$ is the independent variable here.

Worth noting

$\frac{dx}{x}$ belongs to $y = vx$ ; with $x = vy$ the other side is $\frac{dy}{y}$ .

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