Question

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

• A. $\frac{-2}{3}$
• B. $\frac{-1}{3}$
• C. $\frac{4}{3}$
• D. $\frac{1}{3}$

Answer 1

The correct option is D

$\frac{1}{3}$

Explanation of the correct answer:

Step 1: Separating the coefficient as per the variable:

From the equation given in the question,

$(2+\sin x)\left(\frac{dy}{dx}\right)+(y+1)\cos x=0$ (i)

$(2+\sin x)\left(dy\right)=-(y+1)\cos xdx$

$\frac{1}{y+1}dy=-\frac{\cos x}{2+\sin x}dx$ (ii)

Step 2: Integrating on both sides:

By applying integration on both sides we get,

$\int \frac{1}{y+1}dy=-\int \frac{\cos x}{2+\sin x}dx$

$\log (y+1)=-\log (2+\sin x)+\log \left(c\right)$

$\log (y+1)+\log (2+\sin x)=\log \left(c\right)$

$(y+1)(2+\sin x)=c$ (iii)

where $c$ is the constant of integration.

Step 3: Substituting the given value:

It is given that the value of $y\left(0\right)$ is $1$.

Therefore, by substituting the value of $x$ as $0$ and $y$ as $1$, we get the value of $c$ as

$(1+1)(2+\sin (0\left)\right)=c$

$c=2\times 2$

$c=4$

Step 4: Calculating the value of $y\left(\frac{\pi }{2}\right)$:

On substituting the value of $x$ as $\frac{\pi }{2}$ in equation (iii), we get

$(y+1)(2+\sin (\frac{\pi }{2}\left)\right)=4$

$(y+1)(2+1)=4$

$(y+1)=\frac{4}{3}$

$y=\frac{1}{3}$

Therefore the value of $y\left(\frac{\pi }{2}\right)$ turns out to be $\frac{1}{3}$.

Hence, Option (D) is correct.