A homogeneous equation is written as dx/dy = G(x/y). Which substitution, with its derivative, separates it?

x = vy, with dx/dy = v + ydv/dy

Question 14

MediumOriginal question

A homogeneous equation is written as $\frac{dx}{dy} = G\left(\frac{x}{y}\right)$ . Which substitution, with its derivative, separates it?

x = vy

, with

\frac{dx}{dy} = v + y\frac{dv}{dy}

Correct answerYour answer

x = vy

, with

\frac{dx}{dy} = y\frac{dv}{dy}

Correct answerYour answer

x = vy

, with

\frac{dx}{dy} = v

Correct answerYour answer

x = v + y

, with

\frac{dx}{dy} = \frac{dv}{dy}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

Here $x$ is the unknown, so put $x = vy$ and differentiate with respect to $y$ .
Product rule: $\frac{dx}{dy} = v + y\frac{dv}{dy}$ ; then $y\frac{dv}{dy} = G(v) - v$ separates.

Worth noting

$y\frac{dv}{dy}$ alone drops the first term of the product rule, $v \cdot 1$ .

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