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Homogeneous & Linear Differential Equations · Question 42 of 120

Which is the general solution of the equation?

Question 42

MediumOriginal question

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

Which is the general solution of the equation?

e^{x/y} = \log\lvert y \rvert + C

Correct answerYour answer

e^{y/x} = \log\lvert x \rvert + C

Correct answerYour answer

e^{-x/y} = \log\lvert y \rvert + C

Correct answerYour answer

e^{x/y} = \log\lvert x \rvert + C

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

$\int e^{v}\,dv = e^{v}$ and $\int \frac{dy}{y} = \log\lvert y \rvert$ .
Put back $v = \frac{x}{y}$ : $e^{x/y} = \log\lvert y \rvert + C$ .

Worth noting

$e^{y/x} = \log\lvert x \rvert + C$ puts back $v$ as $\frac{y}{x}$ ; with $x = vy$ , $v = \frac{x}{y}$ .

Concept: Solving a homogeneous equation with \(x = vy\)
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsSolving homogeneous equations

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