What is the general solution of y^2dx/dy = x^2 + xy?

Question 47

MediumOriginal question

What is the general solution of $y^{2}\frac{dx}{dy} = x^{2} + xy$ ?

\frac{y}{x} = \log\lvert y \rvert + C

Correct answerYour answer

\frac{x}{y} + \log\lvert y \rvert = C

Correct answerYour answer

\frac{y}{x} + \log\lvert x \rvert = C

Correct answerYour answer

\frac{y}{x} + \log\lvert y \rvert = C

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$x = vy$ : $v + y\frac{dv}{dy} = v^{2} + v$ , so $\frac{dv}{v^{2}} = \frac{dy}{y}$ .
$-\frac{1}{v} = \log\lvert y \rvert + c$ , and $\frac{1}{v} = \frac{y}{x}$ : $\frac{y}{x} + \log\lvert y \rvert = C$ .

Worth noting

$\frac{y}{x} = \log\lvert y \rvert + C$ loses the minus sign of $\int \frac{dv}{v^{2}} = -\frac{1}{v}$ .

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