`y(xy+1)dx+x(1+xy+x^(2)y^(2))dy=0`

Explanation

To solve the differential equation \( y(xy + 1)dx + x(1 + xy + x^2y^2)dy = 0 \), we will follow these steps:

### Step 1: Rewrite the equation
The given equation can be rewritten as:
\[ y(xy + 1)dx + x(1 + xy + x^2y^2)dy = 0 \]

### Step 2: Expand the terms
Expanding the terms gives:
\[ y \cdot xy \, dx + y \, dx + x \, dy + x \cdot xy \, dy + x^2y^2 \, dy = 0 \]
This simplifies to:
\[ xy^2 \, dx + y \, dx + x \, dy + x^2y \, dy + x^2y^2 \, dy = 0 \]

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

### Step 4: Factor out common terms
Factoring out common terms from the grouped equation:
\[ y(xy + 1)dx + x(1 + xy + x^2y^2)dy = 0 \]

### Step 5: Recognize the total differential
We can recognize that:
\[ d(xy) = y \, dx + x \, dy \]
Thus, we can rewrite the equation as:
\[ d(xy) + (x^2y^2)dy = 0 \]

### Step 6: Divide by \(x^2y^2\)
Dividing the entire equation by \(x^2y^2\):
\[ \frac{d(xy)}{x^2y^2} + dy = 0 \]

### Step 7: Substitute \(m = xy\)
Let \(m = xy\). Then, we have:
\[ \frac{dm}{m^2} + dy = 0 \]

### Step 8: Integrate both sides
Integrating both sides:
\[ \int \frac{dm}{m^2} + \int dy = 0 \]
This results in:
\[ -\frac{1}{m} + y = C \]
where \(C\) is the constant of integration.

### Step 9: Substitute back \(m = xy\)
Substituting back \(m = xy\):
\[ -\frac{1}{xy} + y = C \]

### Step 10: Rearranging the equation
Rearranging gives:
\[ -\frac{1}{xy} = C - y \]
or
\[ \frac{1}{xy} = y - C \]

### Final Step: Conclusion
Thus, the solution to the differential equation is:
\[ \frac{1}{xy} + y = C \]