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 $logy=asinx$ ...(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 $\frac{\frac{1}{y}\frac{dy}{dx}}{\log y}=\frac{a\cos x}{a\sin x}$
$\Rightarrow$ $\frac{dy}{dx}=y\log y\cot x$
$\Rightarrow$ $y\log y=\tan x\frac{dy}{dx}$