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

Homogeneous & Linear Differential Equations · Question 52 of 120

Question 52

HardOriginal question

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

x^{2} - y^{2} = Cx

Correct answerYour answer

x^{2} + y^{2} = Cx

Correct answerYour answer

x^{2} - y^{2} = C

Correct answerYour answer

y^{2} = Cx^{2}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

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

Worth noting

$x^{2} - y^{2} = C$ loses the factor $x$ when $1 - v^{2}$ is multiplied out.

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

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