If `y=y(x)` and `(2+sinx)/(y+1) ((dy)/(dx))= - cos x, y(0)=1`, then `y(pi//2)` equals

Options

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

Answer

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 \( x \):

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

Now, we can separate the variables:

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

### 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 substitution. Let \( u = 2 + \sin x \), then \( du = \cos x dx \). This gives us:

\[ \int -\frac{1}{u} du = -\ln |u| + C_2 = -\ln |2 + \sin x| + C_2 \]

Thus, we have:

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

### Step 3: Simplify the Equation

Exponentiating both sides, we get:

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

We can drop the absolute value since \( y + 1 \) will be positive given the initial condition. Therefore, we have:

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

### Step 4: Apply the Initial Condition

Using the initial condition \( y(0) = 1 \):

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

This simplifies to:

\[ 2 = \frac{C}{2} \]

Thus, \( C = 4 \). Now substituting back into our equation, we have:

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

### Step 5: Solve for \( y \)

Rearranging gives us:

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

### Step 6: Find \( y\left(\frac{\pi}{2}\right) \)

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 \( \sin\left(\frac{\pi}{2}\right) = 1 \):

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

Thus, the final answer is:

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