Solution of the differential equation
`(dy)/(dx) +y cot x =2 cos x ` is

Options

A. `y sin x +cos^(2)x=c`
B. `y = sin x +c "cosec"x`
C. `(y-sin x) sin x=c`
D. None

Answer

To solve the differential equation

\[ \frac{dy}{dx} + y \cot x = 2 \cos x, \]

we will follow these steps:

### Step 1: Identify the form of the differential equation
The given equation is in the standard form of a first-order linear differential equation:

\[ \frac{dy}{dx} + P(x) y = Q(x), \]

where \( P(x) = \cot x \) and \( Q(x) = 2 \cos x \).

**Hint:** Recognize the standard form of a linear differential equation to identify \( P(x) \) and \( Q(x) \).

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is given by:

\[ \mu(x) = e^{\int P(x) \, dx} = e^{\int \cot x \, dx}. \]

The integral of \( \cot x \) is \( \ln |\sin x| \), so:

\[ \mu(x) = e^{\ln |\sin x|} = |\sin x|. \]

We can drop the absolute value since \( \sin x \) is positive in the intervals we are considering.

**Hint:** The integrating factor is crucial for solving linear differential equations and is derived from \( P(x) \).

### Step 3: Multiply the entire equation by the integrating factor
Now we multiply the entire differential equation by \( \sin x \):

\[ \sin x \frac{dy}{dx} + y \sin x \cot x = 2 \sin x \cos x. \]

This simplifies to:

\[ \sin x \frac{dy}{dx} + y \frac{\sin^2 x}{\sin x} = 2 \sin x \cos x. \]

**Hint:** Multiplying by the integrating factor allows us to express the left-hand side as the derivative of a product.

### Step 4: Rewrite the left-hand side
The left-hand side can be rewritten as:

\[ \frac{d}{dx}(y \sin x) = 2 \sin x \cos x. \]

**Hint:** Recognize that the left-hand side is the derivative of the product \( y \sin x \).

### Step 5: Integrate both sides
Now we integrate both sides:

\[ \int \frac{d}{dx}(y \sin x) \, dx = \int 2 \sin x \cos x \, dx. \]

The left-hand side simplifies to:

\[ y \sin x = \int 2 \sin x \cos x \, dx. \]

Using the identity \( 2 \sin x \cos x = \sin(2x) \), we have:

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

where \( C \) is the constant of integration.

**Hint:** Use trigonometric identities to simplify the integration.

### Step 6: Solve for \( y \)
Now, we can solve for \( y \):

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

This can be rewritten as:

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

**Hint:** Isolate \( y \) to express it in terms of \( x \) and the constant \( C \).

### Final Solution
The general solution of the differential equation is:

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