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

Options

A. `2y =sin y (1-2cx ^(2))`
B. `2x= cot y (1+2cx^(2))`
C. `2x = sin y (1+ 2cx ^(2))`
D. ` 2x sin y=1-2cx^(2)`

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 can rearrange the equation to isolate the terms involving \(y\) on one side:

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

### Step 2: Dividing by \(\tan y \sin y\)
Next, we divide the entire equation by \(\tan y \sin y\):

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

### Step 3: Substituting Variables
Now, we will make the substitutions:

\[ v = \frac{1}{\sin y} \quad \text{and} \quad u = \frac{1}{x}. \]

This gives us:

\[ \sin y = \frac{1}{v} \quad \text{and} \quad \frac{dy}{dx} = -\frac{1}{\sin^2 y} \cos y \frac{dy}{dx} = -v^2 \cos y \frac{dy}{dx}. \]

### Step 4: Rewrite the Equation
Substituting these into the equation gives:

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

### Step 5: Linear Differential Equation
Now, we can rearrange this into a standard linear form:

\[ \frac{dv}{dx} - \frac{v}{x} = -\frac{1}{x^2}. \]

### Step 6: Finding the Integrating Factor
The integrating factor \( \mu(x) \) is given by:

\[ \mu(x) = e^{\int -\frac{1}{x} dx} = e^{-\log x} = \frac{1}{x}. \]

### Step 7: Multiply by the Integrating Factor
Multiplying the entire equation by the integrating factor:

\[ \frac{1}{x} \frac{dv}{dx} - \frac{v}{x^2} = -\frac{1}{x^3}. \]

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

\[ \int \left(\frac{1}{x} \frac{dv}{dx} - \frac{v}{x^2}\right) dx = \int -\frac{1}{x^3} dx. \]

This simplifies to:

\[ v \cdot \frac{1}{x} = \frac{1}{2x^2} + C. \]

### Step 9: Substitute Back for \(v\)
Now substituting back for \(v\):

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

### Step 10: Rearranging to Find \(y\)
Rearranging gives:

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

Cross-multiplying yields:

\[ 2x - \sin y = 2Cx^2 \sin y. \]

### Step 11: Final Rearrangement
Finally, we can express this as:

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

This is the general solution of the differential equation.