Question

Solve:

$\sin x\frac{dy}{dx}=y\log y$ and $y=e$ when $x=\frac{\pi }{2}$. We get the solution $y=e^{\tan \left(\frac{mx}{k}\right)}$. Then find $k-m$.

Answer 1

$\sin x\frac{dy}{dx}=y\log y$

$\Rightarrow \frac{dy}{dx}=y\text{cosec}x\log y\Rightarrow \frac{\frac{dy}{dx}}{y\log y}=\text{cosec}x$

$\Rightarrow \int \frac{\frac{dy}{dx}}{y\log y}dx=\int \text{cosec}xdx$

$\Rightarrow \log \left(\log y\right)=-\log \left(\cos \frac{x}{2}\right)+\log \left(\sin \frac{x}{2}\right)+c$

$\Rightarrow \log \left(\log c\right)=\log \left(\frac{1}{\sqrt{2}}\right)-\log \left(\frac{1}{\sqrt{2}}\right)+c$

$\Rightarrow c=0$

$\therefore y=e^{\tan \left(\frac{x}{2}\right)}$

Hence, $k-m=1$