Written as dx/dy = G(v), what is G(v)?

Question 32

EasyOriginal question

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

Written as $\frac{dx}{dy} = G(v)$ , what is $G(v)$ ?

v + e^{v}

Correct answerYour answer

v + 1

Correct answerYour answer

v + e^{-v}

Correct answerYour answer

\frac{e^{v}}{ve^{v} + 1}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

$\frac{dx}{dy} = \frac{x\,e^{x/y} + y}{y\,e^{x/y}} = \frac{x}{y} + \frac{y}{y\,e^{x/y}}$ .
So $G(v) = v + e^{-v}$ .

Worth noting

$v + e^{v}$ turns $\frac{1}{e^{v}}$ into $e^{v}$ ; dividing by $e^{v}$ gives $e^{-v}$ .

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

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