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

Answer

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

### Step 1: Rewrite the Differential Equation
We start with the equation:
\[ (x^2 y - x^2)dx + (xy^2 - y^2)dy = 0 \]
This can be rearranged to:
\[ (x^2 y - x^2)dx = -(xy^2 - y^2)dy \]

### Step 2: Factor the Terms
Factoring out common terms, we have:
\[ x^2(y - 1)dx = -y^2(x - 1)dy \]

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

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

#### Left Side Integration:
Using polynomial long division, we can rewrite:
\[ \frac{y^2}{y - 1} = y + 1 + \frac{1}{y - 1} \]
Thus,
\[ \int \left(y + 1 + \frac{1}{y - 1}\right) dy = \frac{y^2}{2} + y + \ln|y - 1| + C_1 \]

#### Right Side Integration:
Similarly, for the right side:
\[ \frac{x^2}{x - 1} = x + 1 + \frac{1}{x - 1} \]
Thus,
\[ -\int \left(x + 1 + \frac{1}{x - 1}\right) dx = -\left(\frac{x^2}{2} + x + \ln|x - 1|\right) + C_2 \]

### Step 5: Combine the Results
Setting the integrals equal gives:
\[ \frac{y^2}{2} + y + \ln|y - 1| = -\left(\frac{x^2}{2} + x + \ln|x - 1|\right) + C \]
where \(C = C_2 - C_1\).

### Step 6: Rearranging the Equation
We can rearrange this to express it in a more standard form:
\[ \frac{y^2}{2} + y + \ln|y - 1| + \frac{x^2}{2} + x + \ln|x - 1| = C \]

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