The solution of the differential equation `y(xy + 2x^2y^2) dx + x(xy-x^2y^2)dy = 0` is given by

Options

A. `2log|x|-log|y|-(1)/(xy)=C`
B. `2log|y|-log|x|-(1)/(xy)=C`
C. `2log|x|+log|y|+(1)/(xy)=C`
D. `2log|y|+log|x|+(1)/(xy)=C`

Explanation

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

### Step 1: Rewrite the Differential Equation
We start with the given differential equation:
\[ y(xy + 2x^2y^2) \, dx + x(xy - x^2y^2) \, dy = 0 \]

### Step 2: Factor Out Common Terms
We can factor out common terms from the equation:
\[ y^2x \, dx + 2x^2y^3 \, dx + xy^2 \, dy - x^3y^2 \, dy = 0 \]
This can be rewritten as:
\[ xy^2 \, dx + 2x^2y^3 \, dx + (x^2y - x^3y^2) \, dy = 0 \]

### Step 3: Group the Terms
Now, we can group the terms:
\[ xy^2 \, dx + x^2y \, dy + 2x^2y^3 \, dx - x^3y^2 \, dy = 0 \]
This can be expressed as:
\[ xy^2 \, dx + x^2y \, dy + 2x^2y^3 \, dx - x^3y^2 \, dy = 0 \]

### Step 4: Factor Further
Now we can factor out \( xy \) and \( x^2y^2 \):
\[ xy \left( y \, dx + x \, dy \right) + x^2y^2 \left( 2y \, dx - x \, dy \right) = 0 \]

### Step 5: Divide by \( x^3y^3 \)
Next, we divide the entire equation by \( x^3y^3 \):
\[ \frac{y \, dx + x \, dy}{x^2y^2} + \frac{2y \, dx - x \, dy}{xy} = 0 \]

### Step 6: Recognize the Derivative
Notice that \( \frac{d(xy)}{dx} = y \, dx + x \, dy \). Thus, we can rewrite the equation as:
\[ \frac{d(xy)}{x^2y^2} + 2 \frac{d(x)}{x} - \frac{d(y)}{y} = 0 \]

### Step 7: Integrate
Now we can integrate both sides:
\[ \int \frac{d(xy)}{(xy)^2} + 2 \int \frac{dx}{x} - \int \frac{dy}{y} = 0 \]
This gives us:
\[ -\frac{1}{xy} + 2 \ln |x| - \ln |y| = C \]

### Step 8: Rearranging the Equation
Rearranging this equation yields:
\[ 2 \ln |x| - \ln |y| - \frac{1}{xy} = C \]

### Final Solution
Thus, the solution of the given differential equation is:
\[ 2 \ln |x| - \ln |y| - \frac{1}{xy} = C \]