Find the particular solution of the differential equation $\frac{dy}{dx} = y \cot 2x$, given that $y\left(\frac{\pi}{4}\right) = 2$.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → $y = 2\sqrt{\sin 2x}$ ##
$\frac{dy}{dx} = y \cot 2x$
Given $y = 2, x = \frac{\pi}{4}$
$\Rightarrow \frac{dy}{y} = \cot 2x \, dx$
$\Rightarrow \int \frac{dy}{y} = \int \cot 2x \, dx$
$\Rightarrow \log y = \frac{1}{2} \log |\sin 2x| + \log C$
$\Rightarrow \log \frac{y}{\sqrt{\sin 2x}} = \log C$
$\Rightarrow \frac{y}{\sqrt{\sin 2x}} = C \quad \dots(i)$
Put $x = \frac{\pi}{4}, y = 2$ in Eq. (i)
$C = 2$
Hence, $y = 2\sqrt{\sin 2x}$