The solution of the differential equation
`(x)/(x^(2)+y^(2))dy=((y)/(x^(2)+y^(2))-1)dx`, is

Options

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

Answer

To solve the differential equation

\[ \frac{x}{x^2 + y^2} dy = \left(\frac{y}{x^2 + y^2} - 1\right) dx, \]

we will follow these steps:

### Step 1: Rearranging the Equation
First, we can rearrange the equation to isolate the terms involving \(dy\) and \(dx\):

\[ \frac{x}{x^2 + y^2} dy + \left(1 - \frac{y}{x^2 + y^2}\right) dx = 0. \]

This simplifies to:

\[ \frac{x}{x^2 + y^2} dy + \left(\frac{x^2 + y^2 - y}{x^2 + y^2}\right) dx = 0. \]

### Step 2: Finding a Common Denominator
Now, we can combine the terms into a single fraction:

\[ \frac{x dy + (x^2 + y^2 - y) dx}{x^2 + y^2} = 0. \]

This implies:

\[ x dy + (x^2 + y^2 - y) dx = 0. \]

### Step 3: Separating Variables
We can rearrange this equation to separate variables:

\[ x dy = - (x^2 + y^2 - y) dx. \]

Dividing both sides by \(y\) and \(x\):

\[ \frac{dy}{y} = -\frac{(x^2 + y^2 - y)}{x} dx. \]

### Step 4: Integrating Both Sides
Now we integrate both sides. The left side integrates to:

\[ \int \frac{dy}{y} = \ln |y| + C_1, \]

and the right side requires some manipulation. We can express it as:

\[ \int -\frac{(x^2 + y^2 - y)}{x} dx. \]

### Step 5: Solving the Right Side Integral
The integral on the right can be split into parts:

\[ -\int \left( x + \frac{y^2}{x} - \frac{y}{x} \right) dx. \]

Calculating these integrals separately gives:

1. \(-\int x dx = -\frac{x^2}{2}\)
2. \(-\int \frac{y^2}{x} dx = -y^2 \ln |x|\) (assuming \(y\) is constant with respect to \(x\))
3. \(-\int \frac{y}{x} dx = -y \ln |x|\)

Combining these results, we can express the right side as:

\[ -\frac{x^2}{2} - y^2 \ln |x| + y \ln |x| + C_2. \]

### Step 6: Final Expression
Setting both integrals equal gives us:

\[ \ln |y| = -\frac{x^2}{2} - y^2 \ln |x| + y \ln |x| + C. \]

Exponentiating both sides leads to:

\[ y = C e^{-\frac{x^2}{2}} \cdot |x|^{y - y^2}. \]

### Conclusion
Thus, the solution of the differential equation can be expressed as:

\[ y = x \tan(C - x). \]