Solve : `(x^2-yx^2)dy+(y^2+x^2y^2)dx=0`

Answer

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

### Step 1: Rewrite the equation
We start with the equation:
\[ (x^2 - yx^2)dy + (y^2 + x^2y^2)dx = 0 \]
We can rearrange it as:
\[ (x^2 - yx^2)dy = -(y^2 + x^2y^2)dx \]

### Step 2: Factor out common terms
From the left-hand side, we can factor out \(x^2\):
\[ x^2(1 - y)dy = -(y^2(1 + x^2))dx \]

### Step 3: Separate variables
Now, we can separate the variables \(y\) and \(x\):
\[ \frac{1 - y}{y^2} dy = -\frac{1 + x^2}{x^2} dx \]

### Step 4: Integrate both sides
Next, we integrate both sides:
\[ \int \frac{1 - y}{y^2} dy = -\int \frac{1 + x^2}{x^2} dx \]

### Step 5: Solve the left integral
The left integral can be split:
\[ \int \left(\frac{1}{y^2} - \frac{1}{y}\right) dy = \int y^{-2} dy - \int y^{-1} dy \]
Calculating these integrals:
\[ \int y^{-2} dy = -\frac{1}{y}, \quad \int y^{-1} dy = \ln |y| \]
Thus,
\[ -\frac{1}{y} - \ln |y| \]

### Step 6: Solve the right integral
The right integral can also be split:
\[ -\int \left(\frac{1}{x^2} + 1\right) dx = -\left(-\frac{1}{x} + x\right) = \frac{1}{x} - x \]

### Step 7: Combine results
Putting it all together, we have:
\[ -\frac{1}{y} - \ln |y| = \frac{1}{x} - x + C \]

### Step 8: Rearranging the equation
Rearranging gives us:
\[ \ln |y| + \frac{1}{y} + x - \frac{1}{x} - C = 0 \]

This is the implicit solution to the differential equation.