Which of these differential equations is homogeneous?

(x^{2} + xy)\,dy = (x^{2} + y^{2})\,dx

Correct answerYour answer

\frac{dy}{dx} = \frac{x + y + 1}{x - y}

Correct answerYour answer

\frac{dy}{dx} = \frac{x^{2} + y}{x}

Correct answerYour answer

\frac{dy}{dx} = x + y^{2}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

Written as $\frac{dy}{dx} = F(x, y)$ , the equation is homogeneous when $F$ is homogeneous of degree 0.
$F = \frac{x^{2} + y^{2}}{x^{2} + xy}$ is a ratio of two degree-2 expressions, so degree 0. The $+1$ , $x^{2} + y$ and $y^{2}$ spoil the others.

Worth noting

The $+1$ in $\frac{x + y + 1}{x - y}$ has degree 0 while $x + y$ has degree 1.

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

Explanation