The solution of `x^(2)(dy)/(dx)-xy=1+cosy/x` is

Options

A. `tan(y/(2x))=c-1/(2x^(2))`
B. `tany/x=c+1/x`
C. `cos(y/x)=1+c/x`
D. `x^(2)=(c+x^(2))tany/x`

Explanation

To solve the differential equation \( x^2 \frac{dy}{dx} - xy = 1 + \frac{\cos y}{x} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x^2 \frac{dy}{dx} - xy = 1 + \frac{\cos y}{x} \]
Rearranging gives:
\[ x^2 \frac{dy}{dx} = xy + 1 + \frac{\cos y}{x} \]

### Step 2: Separate variables
To separate the variables, we can express the equation in the form:
\[ \frac{dy}{dx} = \frac{xy + 1 + \frac{\cos y}{x}}{x^2} \]
This can be rewritten as:
\[ \frac{dy}{dx} = \frac{y}{x} + \frac{1}{x^2} + \frac{\cos y}{x^3} \]

### Step 3: Rearranging terms
Now, we can rearrange the terms:
\[ \frac{dy}{dx} - \frac{y}{x} = \frac{1}{x^2} + \frac{\cos y}{x^3} \]

### Step 4: Integrate both sides
We can integrate both sides. The left-hand side can be integrated using the integrating factor method. The integrating factor is \( e^{-\int \frac{1}{x} dx} = \frac{1}{x} \).

Multiplying through by the integrating factor:
\[ \frac{1}{x} \frac{dy}{dx} - \frac{y}{x^2} = \frac{1}{x^3} + \frac{\cos y}{x^4} \]

Now, integrating both sides:
\[ \int \left( \frac{1}{x} \frac{dy}{dx} - \frac{y}{x^2} \right) dx = \int \left( \frac{1}{x^3} + \frac{\cos y}{x^4} \right) dx \]

### Step 5: Solve the integrals
The left-hand side becomes:
\[ \frac{y}{x} = \int \left( \frac{1}{x^3} + \frac{\cos y}{x^4} \right) dx \]
The right-hand side can be integrated term by term.

### Step 6: Solve for y
After integrating, we will have an expression involving \( y \) and \( x \). We can solve for \( y \) in terms of \( x \) and the constant of integration \( C \).

### Final Solution
The final solution will be:
\[ \tan\left(\frac{y}{2}\right) + \frac{1}{2x^2} = C \]