Which of these differential equations is not homogeneous?

\frac{dy}{dx} = \frac{y}{x} + \sin\frac{y}{x}

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x\,dy - y\,dx = \sqrt{x^{2} + y^{2}}\,dx

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

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

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CorrectNot quiteAnswer D

Correct answer: D

Each equation is homogeneous when its $\frac{dy}{dx} = F(x, y)$ has $F$ of degree 0.
$\frac{x + y}{x^{2}}$ has degree $1 - 2 = -1$ ; the other three right sides have degree 0.

Worth noting

$x\,dy - y\,dx = \sqrt{x^{2} + y^{2}}\,dx$ is homogeneous: $\frac{dy}{dx} = \frac{y + \sqrt{x^{2} + y^{2}}}{x}$ has degree 0.

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

Explanation