Question

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

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

Answer 1

The correct option is B

$y\log y=\tan x\left(\frac{dy}{dx}\right)$

The explanation for the correct answer.

Solve for the family of curves for $y=e^{a\sin x}$

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

Taking$\log$ both sides.

$$\begin{gathered} \log y=\log \left(e^{a\sin \left(x\right)}\right) \\ \log y=a\sin \left(x\right)\log e[\mathbf{\because }\mathit{l}\mathit{o}\mathit{g}\mathbf{}\mathit{e}\mathbf{}\mathbf{=}\mathbf{1}] \end{gathered}$$

$$\begin{gathered} \sin x=\frac{\log y}{a} \\ \therefore \frac{dy}{dx}=e^{a\sin \left(x\right)}\times a\cos \left(x\right) \\ =y\cos \left(x\right).\frac{\log y}{\sin \left(x\right)} \\ \Rightarrow y\log y=\tan \left(x\right)\frac{dy}{dx} \end{gathered}$$

Hence, option(B) is the correct answer.