Question

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

• A. $ \log y=\tan x\frac{dy}{dx} $
• B. $ y\log y=\tan x\frac{dy}{dx} $
• C. $ y\log y=\sin x\frac{dy}{dx} $
• D. $ \log y=\cos x\frac{dy}{dx} $
• E. $ y\log y=\cos x\frac{dy}{dx} $

Answer 1

Answer: (B)

Given curve is $ y={{e}^{a\sin x}} $ ...(i) Taking log on both sides, we get $ log\text{ }y=a\text{ }sin\text{ }x $ ...(ii) Differentiating w.r.t. $ x, $ we get $ \frac{1}{y}\frac{dy}{dx}=a\cos x $ ...(iii) Dividing Eq. (iii) by Eq. (ii), we get $ \frac{\frac{1}{y}\frac{dy}{dx}}{\log y}=\frac{a\cos x}{a\sin x} $
$ \Rightarrow $ $ \frac{dy}{dx}=y\log y\cot x $
$ \Rightarrow $ $ y\log y=\tan x\frac{dy}{dx} $