For $x > 0$ , $\frac{dy}{dx} - \frac{2}{x}y = x$ is multiplied by its integrating factor. Which exact derivative does the left side become?

\frac{d}{dx}\left(x^{2}y\right)

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\frac{d}{dx}\left(\frac{y}{x}\right)

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\frac{d}{dx}\left(\frac{y}{x^{2}}\right)

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\frac{d}{dx}\left(e^{-2x}y\right)

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Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

$P = -\frac{2}{x}$ , so the factor is $e^{-2\log x} = \frac{1}{x^{2}}$ .
By the quotient rule, $\frac{1}{x^{2}}\frac{dy}{dx} - \frac{2}{x^{3}}y = \frac{d}{dx}\left(\frac{y}{x^{2}}\right)$ .

Worth noting

$x^{2}y$ uses the factor for $+\frac{2}{x}$ ; the minus sign turns it into $\frac{1}{x^{2}}$ .

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

For x>0x > 0x>0, dydx−2xy=x\frac{dy}{dx} - \frac{2}{x}y = xdxdy​−x2​y=x is multiplied by its integrating factor. Which exact derivative does the left side become?