If `((2+cosx)/(3+y))(dy)/(dx)+sinx=0` and `y(0)=1`, then `y((pi)/(3))` is equal to

Options

A. `(4)/(3)`
B. `(7)/(3)`
C. `(1)/(3)`
D. 1

Answer

To solve the differential equation given by

\[ \frac{(2 + \cos x)}{(3 + y)} \frac{dy}{dx} + \sin x = 0 \]

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

### Step 1: Rearranging the Equation
We can rearrange the equation to isolate \(\frac{dy}{dx}\):

\[ \frac{(2 + \cos x)}{(3 + y)} \frac{dy}{dx} = -\sin x \]

This leads to:

\[ \frac{dy}{dx} = -\frac{(3 + y) \sin x}{(2 + \cos x)} \]

### Step 2: Separating Variables
Next, we separate the variables \(y\) and \(x\):

\[ \frac{dy}{3 + y} = -\frac{\sin x}{2 + \cos x} dx \]

### Step 3: Integrating Both Sides
Now, we integrate both sides:

\[ \int \frac{dy}{3 + y} = \int -\frac{\sin x}{2 + \cos x} dx \]

The left side integrates to:

\[ \ln |3 + y| \]

For the right side, we can use the substitution \(u = 2 + \cos x\), which gives \(du = -\sin x \, dx\):

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

Thus, we have:

\[ \ln |3 + y| = \ln |2 + \cos x| + C \]

### Step 4: Exponentiating Both Sides
Exponentiating both sides gives:

\[ 3 + y = k(2 + \cos x) \]

where \(k = e^C\).

### Step 5: Applying Initial Condition
Using the initial condition \(y(0) = 1\):

\[ 3 + 1 = k(2 + \cos(0)) \Rightarrow 4 = k(2 + 1) \Rightarrow 4 = 3k \Rightarrow k = \frac{4}{3} \]

### Step 6: Finding the Expression for \(y\)
Substituting \(k\) back into the equation gives:

\[ 3 + y = \frac{4}{3}(2 + \cos x) \]

Thus,

\[ y = \frac{4}{3}(2 + \cos x) - 3 \]

### Step 7: Simplifying the Expression
This simplifies to:

\[ y = \frac{8}{3} + \frac{4}{3} \cos x - 3 = \frac{4}{3} \cos x - \frac{1}{3} \]

### Step 8: Finding \(y\) at \(x = \frac{\pi}{3}\)
Now we need to find \(y\) at \(x = \frac{\pi}{3}\):

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

Since \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\):

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

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

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