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(D) Given the family of lines: $y = mx + \frac{4}{m}$  $(1)$ Differentiating with respect to $x$: $\frac{dy}{dx} = m$ Substituting the value of $m$ in

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$x\left(\frac{dy}{dx}\right)^2 - y\left(\frac{dy}{dx}\right) + 4 = 0$

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(D) Given the family of lines: $y = mx + \frac{4}{m}$  $(1)$ Differentiating with respect to $x$: $\frac{dy}{dx} = m$ Substituting the value of $m$ into equation $(1)$: $y = \left(\frac{dy}{dx}\right)x + \frac{4}{\left(\frac{dy}{dx}\right)}$ Multiplying both sides by $\frac{dy}{dx}$: $y\left(\frac{dy}{dx}\right) = x\left(\frac{dy}{dx}\right)^2 + 4$ Rearranging the terms: $x\left(\frac{dy}{dx}\right)^2 - y\left(\frac{dy}{dx}\right) + 4 = 0$

The differential equation of the family of lines $y = mx + \frac{4}{m}$ obtained by eliminating the arbitrary constant $m$ is

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