Question

Solve the following differential equations :
$\frac{dy}{dx}=\sin x\sin y$

Answer 1

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

$\frac{dy}{\sin y}=\sin xdx$

$cosecydy=\sin xdx$

In $|cosecy+\cot y|+In|C|=\cos x$

In $\left|\frac{C}{cosecy+\cot y}\right|=\cos x$

$C{e}^{-\cos x}=\frac{1}{\sin y}+\frac{\cos y}{\sin y}$

$C{e}^{-\cos x}=\frac{1+\cos y}{\sin y}$

$C{e}^{-\cos x}=\frac{2{\cos }^2\frac{y}{2}}{2\cos \frac{y}{2}\sin \frac{y}{2}}$

$C{e}^{-\cos x}=\cot \left(\frac{y}{2}\right)$

$y=2{\cot }^{-1}\left(C{e}^{-\cos x}\right)$