Question

solve dy/dx,=sinysinx

Answer 1

sin(x-y)=sinx cosy-cosx siny
d/dx siny= cosy dy/dx

Answer 2

$\underline{\texttt{Solution :}}$

$\mathtt{Now,\:\frac{dy}{dx}=siny\:sinx}$

$\implies \mathtt{\frac{dy}{siny}=sinx\:dx}$

$\implies \mathtt{cosecy\:dy=sinx\:dx}$

$\texttt{On integration, we get}$

$\mathtt{\int cosecy\:dy=\int sinx \:dx}$

$\implies \mathtt{log_{e}|tan\frac{y}{2}|=-cosx+C}$

$\texttt{where C is integral constant}$

$\therefore \texttt{the required solution be}$

$\:\:\:\boxed{\mathtt{log_{e}|tan\frac{y}{2}|=-cosx+C}}$

$\underline{\texttt{Integration formulas :}}$

$\mathtt{1.\:\int sinx\:dx=-cosx+C}$

$\texttt{where C is integral constant}$

$\mathtt{2.\:\int cosecx\:dx=log_{e}|tan\frac{x}{2}|+C}$

$\:\:\:\mathtt{=log_{e}|cosecx-cotx|+C}$

$\texttt{where C is integral constant}$