If `y=y(x)` and `((2+sinx)/(y+1))dy/dx=-cosx, y(0)=1`, then `y(pi/2)` equals (A) `1/3` (B) `2/3` (C) `-1/3` (D) `1`

Options

A. `1//3`
B. `2//3`
C. `-1//3`
D. 1

Explanation

To solve the differential equation given by

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

with the initial condition \( y(0) = 1 \), we will follow these steps:

### Step 1: Separate the Variables
We start by rearranging the equation to separate the variables \( y \) and \( x \):

\[ \frac{dy}{y + 1} = -\frac{\cos x}{2 + \sin x} dx \]

### Step 2: Integrate Both Sides
Next, we integrate both sides:

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

The left-hand side integrates to:

\[ \ln |y + 1| + C_1 \]

For the right-hand side, we can use the substitution \( t = 2 + \sin x \), which gives \( dt = \cos x \, dx \). Thus, we have:

\[ -\int \frac{\cos x}{2 + \sin x} dx = -\ln |2 + \sin x| + C_2 \]

### Step 3: Combine the Integrals
Combining the results from both sides, we have:

\[ \ln |y + 1| = -\ln |2 + \sin x| + C \]

where \( C = C_2 - C_1 \).

### Step 4: Exponentiate Both Sides
Exponentiating both sides to eliminate the logarithm gives:

\[ |y + 1| = \frac{K}{2 + \sin x} \]

where \( K = e^C \).

### Step 5: Remove the Absolute Value
Since \( y + 1 \) is positive for our initial condition, we can drop the absolute value:

\[ y + 1 = \frac{K}{2 + \sin x} \]

### Step 6: Solve for \( y \)
Rearranging gives:

\[ y = \frac{K}{2 + \sin x} - 1 \]

### Step 7: Use the Initial Condition
Now we apply the initial condition \( y(0) = 1 \):

\[ 1 = \frac{K}{2 + \sin(0)} - 1 \]

This simplifies to:

\[ 1 = \frac{K}{2} - 1 \implies \frac{K}{2} = 2 \implies K = 4 \]

### Step 8: Substitute \( K \) Back
Substituting \( K \) back into the equation for \( y \):

\[ y = \frac{4}{2 + \sin x} - 1 \]

### Step 9: Find \( y(\frac{\pi}{2}) \)
Now we need to find \( y\left(\frac{\pi}{2}\right) \):

\[ y\left(\frac{\pi}{2}\right) = \frac{4}{2 + \sin\left(\frac{\pi}{2}\right)} - 1 = \frac{4}{2 + 1} - 1 = \frac{4}{3} - 1 = \frac{4}{3} - \frac{3}{3} = \frac{1}{3} \]

### Final Answer
Thus, the value of \( y\left(\frac{\pi}{2}\right) \) is

\[ \boxed{\frac{1}{3}} \]