Q105 / 155 · WBJEE 2023 · Mathematics

The family of curves \(y=e^{a \sin x}\), where ' \(a\) ' is arbitrary constant, is represented by the differential equation

  • Correct answer
    A
    \(y \log y=\tan x \frac{d y}{d x}\)
  • B
    \(y \log x=\cot x \frac{d y}{d x}\)
  • C
    \(\log y=\tan x \frac{d y}{d x}\)
  • D
    \(\log y=\cot x \frac{d y}{d x}\)

Solution

\(\begin{aligned} & \Rightarrow \log y=a \sin x \\ & \Rightarrow \frac{d}{d x}\left(\frac{\log y}{\sin x}\right)=\frac{d}{d x}(a) \Rightarrow \frac{\sin x\left(\frac{d y}{d x}\right)}{y}-\log y \times \cos x=0 \\ & \Rightarrow y \log y=\tan x \times \frac{d y}{d x}\end{aligned}\)

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

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