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 given 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 rewrite the equation by taking the common denominator on the right side:

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

This simplifies to:

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

### Step 2: Cross Multiplying
Now, we can cross-multiply to eliminate the denominators:

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

### Step 3: Rearranging Terms
Next, we will rearrange the equation to isolate terms involving \(y\) and \(x\):

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

### Step 4: Dividing by \(x^2 + y^2\)
Now, divide both sides by \(x^2 + y^2\):

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

### Step 5: Integrating Both Sides
Integrate both sides:

\[ \int \frac{x \, dy + y \, dx}{x^2 + y^2} = \int dx. \]

The left-hand side can be recognized as the derivative of \(\tan^{-1}\left(\frac{y}{x}\right)\):

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

### Step 6: Solving for \(y\)
From the equation, we can express \(y\) in terms of \(x\):

\[ \frac{y}{x} = \tan(x + C). \]

Thus, we have:

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

### Final Solution
The solution to the differential equation is:

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