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Q.
If $\frac{dy}{dx}+cot\,x\cdot y=cos\,x$, $y\left(\frac{\pi}{2}\right)=\frac{1}{2}$, then $y\left(\frac{\pi}{6}\right)=$
Differential Equations
Solution:
It is linear differential equation with $I$.$F.$
$= \text{exp} \int cot\,x\,dx=sin\,x$
$\therefore $ Solution is, $y \,sin \,x =\int sin \,x\,cos\,x\,dx=\frac{sin^{2}\,x}{2}+c$
$x=\frac{\pi}{2}$, $y=\frac{1}{2}$
$\Rightarrow c=0$
$\therefore y=\frac{1}{2} sin\,x$
$\Rightarrow y\left(\frac{\pi}{6}\right)=\frac{1}{4}$.