The solution of the differential equation `(1+xy)xdy+(1-xy)ydx=0` ,is

Options

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

Explanation

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

### Step 1: Rewrite the differential equation
We start with the given equation:
\[ (1 + xy)x dy + (1 - xy)y dx = 0 \]

### Step 2: Expand the equation
Expanding the terms gives:
\[ x dy + xy^2 dy + y dx - xy^2 dx = 0 \]
Rearranging the terms, we have:
\[ x dy + y dx + xy^2 dy - xy^2 dx = 0 \]

### Step 3: Group the terms
We can group the terms involving \(dy\) and \(dx\):
\[ (x + xy^2) dy + (y - xy^2) dx = 0 \]

### Step 4: Isolate the differentials
Rearranging gives:
\[ (x + xy^2) dy = - (y - xy^2) dx \]
This implies:
\[ \frac{dy}{dx} = -\frac{y - xy^2}{x + xy^2} \]

### Step 5: Simplify the expression
We can simplify this to:
\[ \frac{dy}{dx} = -\frac{y(1 - xy)}{x(1 + xy)} \]

### Step 6: Separate variables
We can separate the variables:
\[ \frac{dy}{y(1 - xy)} = -\frac{dx}{x(1 + xy)} \]

### Step 7: Integrate both sides
Now we integrate both sides:
\[ \int \frac{1}{y(1 - xy)} dy = -\int \frac{1}{x(1 + xy)} dx \]

### Step 8: Solve the integrals
Using partial fractions, we can solve the integrals. The left side can be integrated as:
\[ \int \left( \frac{A}{y} + \frac{B}{1 - xy} \right) dy \]
And the right side similarly.

### Step 9: Combine results
After integrating, we combine the results to form the general solution:
\[ -\frac{1}{xy} + \log\left(\frac{y}{x}\right) = C \]

### Step 10: Rearranging the equation
Finally, we can rearrange the equation to get the solution in the form:
\[ -\frac{1}{xy} + \log\left(\frac{y}{x}\right) = C \]

### Final Answer
The solution of the differential equation is:
\[ -\frac{1}{xy} + \log\left(\frac{y}{x}\right) = C \]

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