What is the solution of the differential equation
`xdy- ydx = xy^(2) dx` ?

Options

A. `y+x^(-2) =c`
B. ` y^(2) +2x^(-1) =c`
C. `y=x^(-1) =c`
D. `x^(2) +2xy^(-1) =c`

Explanation

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

### Step 1: Rearranging the Equation
Start by rearranging the given equation:
\[ x \, dy - y \, dx = xy^2 \, dx \]
This can be rewritten as:
\[ x \, dy = y \, dx + xy^2 \, dx \]
Factoring out \( dx \) from the right side gives:
\[ x \, dy = (y + xy^2) \, dx \]

### Step 2: Dividing by \( y^2 \)
Next, we can divide both sides by \( y^2 \):
\[ \frac{x \, dy}{y^2} = \left( \frac{y}{y^2} + \frac{xy^2}{y^2} \right) \, dx \]
This simplifies to:
\[ \frac{x \, dy}{y^2} = \left( \frac{1}{y} + x \right) \, dx \]

### Step 3: Rearranging into a Standard Form
Rearranging gives:
\[ \frac{dy}{y^2} = \left( \frac{1}{x} + 1 \right) \frac{dx}{x} \]

### Step 4: Integrating Both Sides
Now we can integrate both sides:
\[ \int \frac{dy}{y^2} = \int \left( \frac{1}{x} + 1 \right) \frac{dx}{x} \]
The left side integrates to:
\[ -\frac{1}{y} \]
The right side can be split and integrated:
\[ \int \frac{1}{x} \, dx + \int 1 \, dx = \ln |x| + x + C \]

### Step 5: Combining Results
Combining the results gives:
\[ -\frac{1}{y} = \ln |x| + x + C \]

### Step 6: Rearranging to Find y
Rearranging to express \( y \) in terms of \( x \):
\[ y = -\frac{1}{\ln |x| + x + C} \]

### Final Solution
Thus, the solution to the differential equation is:
\[ y = -\frac{1}{\ln |x| + x + C} \]