The solution of the equation
`(x^(2)-yx^(2))dy +(y^(2)+x^(2)y^(2))dx=0`

Options

A. `(1)/(x) +y+log y =c `
B. `(1)/(x) +y+(1)/(y^(2)) =c`
C. `(1)/(x)+(1)/(y) +log y -x=c`
D. None of these

Answer

To solve the differential equation

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

we can follow these steps:

### Step 1: Rearranging the Equation
We can rewrite the equation as:

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

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

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

### Step 3: Separating 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: Simplifying the Left Side
The left side can be simplified:

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

Thus, we rewrite the equation as:

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

### Step 5: Integrating Both Sides
Now we integrate both sides:

\[ \int \left(-\frac{1}{y^2} + \frac{1}{y}\right) dy = \int \left(\frac{1}{x^2} + 1\right) dx. \]

The left side integrates to:

\[ \int \left(-\frac{1}{y^2}\right) dy + \int \frac{1}{y} dy = \frac{1}{y} - \log |y|. \]

The right side integrates to:

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

### Step 6: Combining Results
Combining the results from both sides gives us:

\[ \frac{1}{y} - \log |y| = -\frac{1}{x} + x + C, \]

where \(C\) is the constant of integration.

### Step 7: Rearranging the Equation
Finally, we can rearrange this equation to express it in a more standard form:

\[ \frac{1}{y} + \log |y| + \frac{1}{x} - x = C. \]

This is the general solution of the given differential equation.