Find the general solution of each of the following differential equations:
`x(x^(2)-x^(2)y^(2))dy +y(y^(2)+x^(2)y^(2))dx =0`

Explanation

To find the general solution of the differential equation

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

we can start by rearranging the equation for clarity.

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

\[ x(x^{2}(1 - y^{2}))dy + y(y^{2}(1 + x^{2}))dx = 0. \]

### Step 2: Separating Variables
Now, we can separate the variables \(x\) and \(y\). We can divide both sides by \(x^{2}y^{2}\) (assuming \(x \neq 0\) and \(y \neq 0\)):

\[ \frac{(1 - y^{2})}{y^{2}} dy + \frac{(1 + x^{2})}{x^{2}} dx = 0. \]

### Step 3: Integrating Both Sides
Now we can integrate both sides:

\[ \int \frac{(1 - y^{2})}{y^{2}} dy + \int \frac{(1 + x^{2})}{x^{2}} dx = 0. \]

### Step 4: Performing the Integrals
Let's compute the integrals separately.

1. For the first integral:

\[ \int \left(\frac{1}{y^{2}} - 1\right) dy = \int \frac{1}{y^{2}} dy - \int 1 dy = -\frac{1}{y} - y + C_1. \]

2. For the second integral:

\[ \int \left(\frac{1}{x^{2}} + 1\right) dx = \int \frac{1}{x^{2}} dx + \int 1 dx = -\frac{1}{x} + x + C_2. \]

### Step 5: Combining the Results
Combining the results from both integrals, we have:

\[ -\frac{1}{y} - y + C_1 + \left(-\frac{1}{x} + x + C_2\right) = 0. \]

### Step 6: Simplifying the Equation
We can simplify this to:

\[ -\frac{1}{y} - y - \frac{1}{x} + x + C = 0, \]

where \(C = C_1 + C_2\).

### Step 7: Rearranging the Final Form
Rearranging gives us:

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

### Final General Solution
Thus, the general solution of the differential equation is:

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