Solve `(dy)/(dx)+ycotx=sinx`

Explanation

To solve the differential equation

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

we will follow these steps:

### Step 1: Identify \( p \) and \( q \)

The given equation is in the standard linear form:

\[ \frac{dy}{dx} + p y = q, \]

where \( p = \cot x \) and \( q = \sin x \).

**Hint:** Identify \( p \) and \( q \) from the standard form of a linear differential equation.

### Step 2: Find the integrating factor

The integrating factor \( \mu(x) \) is given by:

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

The integral of \( \cot x \) is:

\[ \int \cot x \, dx = \log(\sin x). \]

Thus, the integrating factor becomes:

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

**Hint:** Use the formula for the integrating factor and remember that the integral of \( \cot x \) is \( \log(\sin x) \).

### Step 3: Multiply the entire equation by the integrating factor

Now, multiply the entire differential equation by \( \sin x \):

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

This simplifies to:

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

**Hint:** Multiplying by the integrating factor transforms the left side into a derivative of a product.

### Step 4: Rewrite the left side as a derivative

The left side can be rewritten as:

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

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

### Step 5: Integrate both sides

Integrate both sides with respect to \( x \):

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

The left side simplifies to:

\[ y \sin x. \]

For the right side, we can use the identity \( \sin^2 x = \frac{1 - \cos(2x)}{2} \):

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

Thus, we have:

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

**Hint:** Use trigonometric identities to simplify the integration of \( \sin^2 x \).

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

Now, isolate \( y \):

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

This can be simplified further:

\[ y = \frac{x}{2 \sin x} - \frac{\sin(2x)}{4 \sin x} + \frac{C}{\sin x}. \]

**Hint:** Rearranging the equation helps to express \( y \) in terms of \( x \).

### Final Solution

The solution to the differential equation is:

\[ y = \frac{x}{2 \sin x} - \frac{\sin(2x)}{4 \sin x} + \frac{C}{\sin x}. \]