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  4. The solution of the differential equation (d y/d x)+y cot x= sin x is y sin x=k(2 x- sin 2 x)+c then k is

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Q. The solution of the differential equation $\frac{d y}{d x}+y \cot x=\sin x$ is $y \sin x=k(2 x-\sin 2 x)+c$ then $k$ is

Differential Equations

Solution:

I.F. $=e^{\int \cot x d x}=e^{i n \sin x}=\sin x$
solution is,
$y \sin x=\int \sin ^2 x d x+c$
$y \sin x=\frac{1}{2} \int(1-\cos 2 x) d x+c$
$y \sin x=\frac{1}{4}(2 x-\sin 2 x)+c$