What is the general solution of dy/dx = y/x + x/y?

Question 51

HardOriginal question

What is the general solution of $\frac{dy}{dx} = \frac{y}{x} + \frac{x}{y}$ ?

y^{2} = x^{2}\log\lvert x \rvert + Cx^{2}

Correct answerYour answer

y^{2} = 2\log\lvert x \rvert + C

Correct answerYour answer

y = 2x\log\lvert x \rvert + Cx

Correct answerYour answer

y^{2} = 2x^{2}\log\lvert x \rvert + Cx^{2}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$y = vx$ : $v + x\frac{dv}{dx} = v + \frac{1}{v}$ , so $v\,dv = \frac{dx}{x}$ .
$\frac{v^{2}}{2} = \log\lvert x \rvert + c$ , so $y^{2} = 2x^{2}\log\lvert x \rvert + Cx^{2}$ .

Worth noting

$y^{2} = x^{2}\log\lvert x \rvert + Cx^{2}$ drops the 2 from $\frac{v^{2}}{2}$ .

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