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

Explanation

To solve the differential equation

\[ (x^2 - y x^2) dy + (y^2 + x^2 y^2) dx = 0, \]

we will follow these steps:

### Step 1: Rearranging the Equation
We start by rewriting the equation in a more manageable form:

\[ (x^2 - y x^2) dy = - (y^2 + x^2 y^2) dx. \]

### Step 2: Factoring Common Terms
Next, we factor out common terms from both sides:

\[ x^2(1 - y) dy = -y^2(1 + x^2) dx. \]

### Step 3: Separating Variables
We can now separate the variables \(y\) and \(x\):

\[ \frac{1 - y}{y^2} dy = -\frac{1 + x^2}{x^2} dx. \]

### Step 4: Integrating Both Sides
Now, we will integrate both sides.

**Left Side:**
\[ \int \left(\frac{1}{y^2} - \frac{1}{y}\right) dy = \int \left(y^{-2} - y^{-1}\right) dy. \]

Calculating the integral:
\[ \int y^{-2} dy = -\frac{1}{y}, \quad \int y^{-1} dy = \ln |y|. \]

Thus, the left side becomes:
\[ -\frac{1}{y} - \ln |y|. \]

**Right Side:**
\[ -\int \left(\frac{1}{x^2} + 1\right) dx = -\left(-\frac{1}{x} + x\right). \]

Thus, the right side becomes:
\[ \frac{1}{x} - x. \]

### Step 5: Combining Results
Combining both sides, we have:
\[ -\frac{1}{y} - \ln |y| = \frac{1}{x} - x + C, \]
where \(C\) is the constant of integration.

### Final Solution
Rearranging gives us the final implicit solution:
\[ \frac{1}{y} + \ln |y| + x - \frac{1}{x} = C. \]

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