If $y = y(x)$ and $\frac{2 + \sin{\ x}}{y + 1}\ \left( \frac{dy}{dx} \right) = - \cos x,\ y(0) = 1$, then $y\left( \frac{\pi}{2} \right)$ equals

Question

Dr. Abubacker · Accepted Answer

(a)
Given, $\frac{dy}{y + 1} = \frac{{- cos}x}{2 + \sin x}\ dx$
$\Rightarrow$ $\int_{}^{}\frac{dy}{y + 1} = - \frac{\int_{}^{}{\cos x}}{2 + \sin x}dx$
$\Rightarrow$ $\log{(y + 1)} = - \log{(2 + \sin x)} + \log c$
When $x = 0,\ y = 1$
$\Rightarrow$ $c = 4$
$	herefore$ $y + 1 = \frac{4}{2 + \sin x}$
At $x = \frac{\pi}{2}$, $y + 1 = \frac{4}{2 + 1}$
$\Rightarrow$ $y = \frac{1}{3}$

Q. If $y = y(x)$ and $\frac{2 + \sin{\ x}}{y + 1}\ \left( \frac{dy}{dx} \right) = - \cos x,\ y(0) = 1$, then $y\left( \frac{\pi}{2} \right)$ equals

Explanation

Reviewed by Dr. Abubacker

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