If `(2+sinx)(dy)/(dx)+(y+1)cosx=0` and `y(0)=1`, then `y((pi)/(2))` is equal to

Options

A. `4/3`
B. `1/3`
C. `-2/3`
D. `-1/3`

Explanation

To solve the differential equation \((2 + \sin x) \frac{dy}{dx} + (y + 1) \cos x = 0\) with the initial condition \(y(0) = 1\), we will follow these steps:

### Step 1: Rewrite the Differential Equation
We start with the given equation:
\[ (2 + \sin x) \frac{dy}{dx} + (y + 1) \cos x = 0 \]
Rearranging gives:
\[ \frac{dy}{dx} = -\frac{(y + 1) \cos x}{(2 + \sin x)} \]

### Step 2: Separate Variables
We can separate the variables \(y\) and \(x\):
\[ \frac{dy}{y + 1} = -\frac{\cos x}{2 + \sin x} dx \]

### Step 3: Integrate Both Sides
Now we integrate both sides:
\[ \int \frac{dy}{y + 1} = -\int \frac{\cos x}{2 + \sin x} dx \]

The left side integrates to:
\[ \ln |y + 1| \]

For the right side, we can use substitution. Let \(u = 2 + \sin x\), then \(du = \cos x \, dx\). The integral becomes:
\[ -\int \frac{1}{u} du = -\ln |u| = -\ln |2 + \sin x| \]

Thus, we have:
\[ \ln |y + 1| = -\ln |2 + \sin x| + C \]

### Step 4: Exponentiate to Solve for \(y\)
Exponentiating both sides gives:
\[ |y + 1| = \frac{C}{2 + \sin x} \]
Since \(y + 1\) can be positive or negative, we can drop the absolute value and write:
\[ y + 1 = \frac{C}{2 + \sin x} \]
Thus:
\[ y = \frac{C}{2 + \sin x} - 1 \]

### Step 5: Use Initial Condition to Find \(C\)
We use the initial condition \(y(0) = 1\):
\[ 1 = \frac{C}{2 + \sin(0)} - 1 \]
This simplifies to:
\[ 1 = \frac{C}{2} - 1 \]
Adding 1 to both sides:
\[ 2 = \frac{C}{2} \]
Multiplying by 2 gives:
\[ C = 4 \]

### Step 6: Substitute \(C\) Back into the Equation
Now substituting \(C\) back into the equation for \(y\):
\[ y = \frac{4}{2 + \sin x} - 1 \]

### Step 7: Find \(y\) at \(x = \frac{\pi}{2}\)
Now we need to find \(y\left(\frac{\pi}{2}\right)\):
\[ y\left(\frac{\pi}{2}\right) = \frac{4}{2 + \sin\left(\frac{\pi}{2}\right)} - 1 \]
Since \(\sin\left(\frac{\pi}{2}\right) = 1\):
\[ y\left(\frac{\pi}{2}\right) = \frac{4}{2 + 1} - 1 = \frac{4}{3} - 1 = \frac{4}{3} - \frac{3}{3} = \frac{1}{3} \]

### Final Answer
Thus, the value of \(y\left(\frac{\pi}{2}\right)\) is:
\[ \boxed{\frac{1}{3}} \]