Question

Solve: $\sin x\frac{dy}{dx}=y\log y$.
Also find the particular solution when $x=\frac{\pi }{2},y=1$

Answer 1

Given, $\sin x\frac{dy}{dx}=y\log y$
$\int \frac{1}{y\log y}dy=\int \frac{1}{\sin x}dx$
$\int \frac{1}{y\log y}dy=\int cosecxdx$
$\log |\log y|=\log |cosecx-\cot x|+\log c$
$\log \left|\frac{\log y}{cosecx-\cot x}\right|=\log c$
$\left|\frac{\log y}{cosecx-\cot x}\right|=c$ ... (i) ....$c\to$ constant of integration
When $x=\frac{\pi }{2},y=1$
$\left|\left(\frac{\log 1}{cosec\frac{\pi }{2}-\cot \frac{\pi }{2}}\right)\right|=c\Rightarrow c=0$
From (i), we have

$\frac{\log y}{1-0}=0$
$\Rightarrow \log y=0$
$\Rightarrow \log y=\log 1$
$\Rightarrow y=1$