Solve:
`xdy+ydx= (xdy-ydx)/(x^(2)+y^(2))`

Answer

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

### Step 1: Rewrite the equation
We start with the equation:
\[ x \, dy + y \, dx = \frac{x \, dy - y \, dx}{x^2 + y^2} \]
To simplify, we can multiply both sides by \( x^2 + y^2 \):
\[ (x^2 + y^2)(x \, dy + y \, dx) = x \, dy - y \, dx \]

### Step 2: Expand and rearrange
Expanding the left-hand side gives:
\[ x^2 \, dy + y^2 \, dy + x^2 \, dx + y^2 \, dx = x \, dy - y \, dx \]
Rearranging terms, we get:
\[ x^2 \, dy + y^2 \, dy + x^2 \, dx + y^2 \, dx - x \, dy + y \, dx = 0 \]

### Step 3: Combine like terms
Combining the terms yields:
\[ (x^2 + y^2) \, dy + (x^2 + y^2) \, dx = 0 \]
This can be factored as:
\[ (x^2 + y^2)(dy + dx) = 0 \]

### Step 4: Solve the equation
Since \( x^2 + y^2 \neq 0 \) for all \( x \) and \( y \) (except at the origin), we can divide by \( x^2 + y^2 \):
\[ dy + dx = 0 \]
This implies:
\[ dy = -dx \]

### Step 5: Integrate both sides
Integrating both sides gives:
\[ \int dy = -\int dx \]
This results in:
\[ y = -x + C \]
where \( C \) is the constant of integration.

### Final Solution
Thus, the solution to the differential equation is:
\[ y = -x + C \]

---