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Homogeneous & Linear Differential Equations · Question 45 of 120

Which separated form does x = vy give?

Question 45

MediumOriginal question

Consider the differential equation $y\,e^{x/y}\,dx = (x\,e^{x/y} + y)\,dy$ , and put $x = vy$ .

Which separated form does $x = vy$ give?

e^{-v}\,dv = \frac{dy}{y}

Correct answerYour answer

\frac{dv}{v + e^{-v}} = \frac{dy}{y}

Correct answerYour answer

e^{v}\,dv = \frac{dx}{x}

Correct answerYour answer

e^{v}\,dv = \frac{dy}{y}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$\frac{dx}{dy} = v + y\frac{dv}{dy} = v + e^{-v}$ , so $y\frac{dv}{dy} = e^{-v}$ .
Separate: $\frac{dv}{e^{-v}} = e^{v}\,dv = \frac{dy}{y}$ .

Worth noting

$e^{-v}\,dv$ multiplies by $e^{-v}$ where the step divides by it.

Concept: Separating after \(x = vy\)
Typical timeTypical: 120 s
Your timeYours: —

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