If $y = \frac{x}{\log | cx |}$ where c is an arbitrary constant is the general solution of the differential equation $\frac{dy}{dx} = (\frac{y}{x}) + ϕ (\frac{y}{x})$ then the function $ϕ (\frac{x}{y})$ is . . . .

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$- \frac{y^{2}}{x^{2}}$

$\frac{y^{2}}{x^{2}}$

$- \frac{x^{2}}{y^{2}}$

$\frac{x^{2}}{y^{2}}$

answer is D.

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Detailed Solution

$y = \frac{x}{\log | c x |} \Rightarrow \log | cx | = \frac{x}{y}$ ;

$\log c + \log x = \frac{x}{y} \Rightarrow \frac{1}{x} = \frac{y - x \frac{d y}{d x}}{y^{2}} \Rightarrow \frac{y^{2}}{x} = y - x \frac{dy}{dx}$ ;

$x \frac{dy}{dx} = y - \frac{y^{2}}{x} \Rightarrow \frac{dy}{dx} = \frac{y}{x} - \frac{y^{2}}{x^{2}}$

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