Q51 / 155 · WBJEE 2023 · Mathematics

If \(y=\frac{x}{\log _e|c x|}\) is the solution of the differential equation \(\frac{d y}{d x}=\frac{y}{x}+\phi\left(\frac{x}{y}\right)\), then \(\phi\left(\frac{x}{y}\right)\) is given by

  • A
    \(\frac{y^2}{x^2}\)
  • Correct answer
    B
    \(-\frac{y^2}{x^2}\)
  • C
    \(\frac{x^2}{y^2}\)
  • D
    \(-\frac{x^2}{y^2}\)

Solution

Hint : \(y=\frac{x}{\ell n|c x|}\) \(\begin{aligned} & \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\ell \mathrm{n}|c x|-1}{(\ell \mathrm{n}|\mathrm{cx}|)^2}=\frac{1}{\ell \mathrm{n}|\mathrm{cx}|}-\frac{1}{(\ell \mathrm{n}|\mathrm{cx}|)^2} \\ & \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}}{\mathrm{x}}-\left(\frac{\mathrm{y}}{\mathrm{x}}\right)^2 \quad \therefore \phi\left(\frac{\mathrm{x}}{\mathrm{y}}\right)=-\frac{\mathrm{y}^2}{\mathrm{x}^2} \end{aligned}\)

Hence, the correct answer is option: 'B'.

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