`(1 + xy) ydx + (1-xy)xdy = 0`

Options

A. `(x)/(y)+(1)/(xy)=k`
B. `log((x)/(y))=(1)/(xy)+k`
C. `(x)/(y)+(1)/(xy)=k`
D. `log((x)/(y))= xy + k`

Explanation

To solve the differential equation \((1 + xy) y \, dx + (1 - xy) x \, dy = 0\), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ (1 + xy) y \, dx + (1 - xy) x \, dy = 0 \]
We can rearrange this to isolate \(dx\) and \(dy\):
\[ (1 + xy) y \, dx = - (1 - xy) x \, dy \]

### Step 2: Dividing by \(x^2y^2\)
Next, we divide the entire equation by \(x^2y^2\):
\[ \frac{(1 + xy) y}{x^2y^2} \, dx + \frac{-(1 - xy) x}{x^2y^2} \, dy = 0 \]
This simplifies to:
\[ \frac{(1 + xy)}{x^2y} \, dx - \frac{(1 - xy)}{y^2} \, dy = 0 \]

### Step 3: Identifying the Exact Form
We can rewrite the equation in a more manageable form:
\[ \frac{y \, dx}{x^2y^2} + \frac{x \, dy}{x^2y^2} = 0 \]
This can be expressed as:
\[ y \, dx + x \, dy = 0 \]

### Step 4: Using the Total Differential
We recognize that:
\[ d(xy) = y \, dx + x \, dy \]
Thus, we can rewrite our equation as:
\[ d(xy) = 0 \]

### Step 5: Integrating the Equation
Integrating both sides gives:
\[ xy = C \]
where \(C\) is a constant.

### Final Solution
The solution to the differential equation is:
\[ xy = C \]