Question

The family of curves y=e^{asin x}, where a is an arbitrary constant, is represented by the differential equation

• A. ylog y=tanx \(\frac{dy}{dx}\)
• B. ylog x=cotx \(\frac{dy}{dx}\)
• C. log y=tanx \(\frac{dy}{dx}\)
• D. log y=cotx \(\frac{dy}{dx}\)

Answer 1

The Correct Option is A

Given: $y = e^{a \sin x}$

Take logarithm on both sides: 

$\Rightarrow \log y = a \sin x \quad \cdots (i)$

Differentiating both sides with respect to $x$:

$\frac{1}{y} \cdot \frac{dy}{dx} = a \cos x$

$\Rightarrow a = \frac{1}{y \cos x} \cdot \frac{dy}{dx}$

Substitute this value of $a$ into equation (i):

$\log y = \left(\frac{1}{y \cos x} \cdot \frac{dy}{dx} \right) \sin x$

$\Rightarrow y \log y = \tan x \cdot \frac{dy}{dx}$

Hence, the correct answer is: $y \log y = \tan x \cdot \frac{dy}{dx}$