Question

If $\frac{dy}{dx}+cotx\cdot y=cosx$, $y\left(\frac{\pi }{2}\right)=\frac{1}{2}$, then $y\left(\frac{\pi }{6}\right)=$

• A. $\frac{1}{\sqrt{2}}$
• B. $\frac{1}{2}$
• C. $\frac{1}{2\sqrt{2}}$
• D. $\frac{1}{4}$

Answer 1

Answer: (D)

It is linear differential equation with $I$.$F.$
$=\text{exp}\int cotxdx=sinx$
$\therefore$ Solution is, $ysinx=\int sinxcosxdx=\frac{si{n}^{2}x}{2}+c$
$x=\frac{\pi }{2}$, $y=\frac{1}{2}$
$\Rightarrow c=0$
$\therefore y=\frac{1}{2}sinx$
$\Rightarrow y\left(\frac{\pi }{6}\right)=\frac{1}{4}$.