For x^2dy/dx = y^2 + 2xy, which of the following statements are true? I. y = vx gives xdv/dx = v^2 + v. II. The variables separate as dv/(v^2 + 2v) = dx/x. III. The general solution is y = Cx(x + y). IV. The line y = x is a solution.

Question 43

MediumOriginal question

For $x^{2}\frac{dy}{dx} = y^{2} + 2xy$ , which of the following statements are true?
I. $y = vx$ gives $x\frac{dv}{dx} = v^{2} + v$ .
II. The variables separate as $\frac{dv}{v^{2} + 2v} = \frac{dx}{x}$ .
III. The general solution is $y = Cx(x + y)$ .
IV. The line $y = x$ is a solution.

I and III onlyCorrect answerYour answerI, II and III onlyCorrect answerYour answerI and II onlyCorrect answerYour answerIII and IV onlyCorrect answerYour answer

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CorrectNot quiteAnswer A

Correct answer: A

I: $v + x\frac{dv}{dx} = v^{2} + 2v$ , so $x\frac{dv}{dx} = v^{2} + v$ ; II forgets to take away $v$ .
III: $\frac{dv}{v(v + 1)} = \frac{dx}{x}$ gives $\frac{v}{v + 1} = Cx$ , that is $\frac{y}{x + y} = Cx$ . IV: on $y = x$ the left side is $x^{2}$ and the right side $3x^{2}$ , so it fails.

Worth noting

II divides by $F(v) = v^{2} + 2v$ instead of by $F(v) - v$ .

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