The Solution of the differential equation `(dy)/(dx) +(1)/(x)tan y =(1)/(x^(2)) tan y sin y` is

Options

A. `2x=sin y (1+2cx^(2))`
B. `2x=sin y(1+cx^(2))`
C. `2x+siny (1+cx^(2))=0`
D. none of these

Answer

To solve the differential equation

\[ \frac{dy}{dx} + \frac{1}{x} \tan y = \frac{1}{x^2} \tan y \sin y, \]

we will follow these steps:

### Step 1: Rearranging the Equation
First, we will rearrange the equation by moving the term \(\frac{1}{x^2} \tan y \sin y\) to the left side:

\[ \frac{dy}{dx} + \frac{1}{x} \tan y - \frac{1}{x^2} \tan y \sin y = 0. \]

### Step 2: Factoring Out \(\tan y\)
Next, we can factor out \(\tan y\) from the terms involving it:

\[ \frac{dy}{dx} + \frac{\tan y}{x} \left(1 - \frac{\sin y}{x}\right) = 0. \]

### Step 3: Substituting \(t = \csc y\)
To simplify the equation, we can use the substitution \(t = \csc y\). Then, we have:

\[ \frac{dy}{dx} = -\csc y \cot y \frac{dy}{dx}. \]

Substituting this into the equation gives:

\[ -\csc y \cot y \frac{dy}{dx} + \frac{\cot y}{x} - \frac{\cot y \sin y}{x^2} = 0. \]

### Step 4: Simplifying the Equation
This simplifies to:

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

### Step 5: Converting to Linear Form
Now, we can express this in a linear form. We can write it as:

\[ \frac{dy}{dx} + \frac{1}{x} \cot y = \frac{1}{x^2} \cot y \sin y. \]

### Step 6: Finding the Integrating Factor
The integrating factor \(I\) can be found using:

\[ I = e^{\int P(x) dx} = e^{\int \frac{1}{x} dx} = e^{\ln |x|} = |x|. \]

### Step 7: Multiplying by the Integrating Factor
Now, we multiply the entire equation by the integrating factor \(x\):

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

### Step 8: Integrating Both Sides
Integrating both sides gives:

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

### Step 9: Solving the Integral
The left side integrates to \(xy\) and the right side can be integrated with respect to \(x\):

\[ xy = -\frac{1}{2} \frac{1}{x^2} + C. \]

### Step 10: Rearranging for \(y\)
Finally, we can rearrange this to express \(y\):

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

### Final Solution
Thus, the solution to the differential equation is:

\[ 2x \sin y = 1 + 2Cx^2. \]