Solve `ydx-xdy=x^(2)ydx`.

Explanation

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

### Step 1: Rearranging the Equation
We start with the given equation:

\[ y \, dx - x \, dy = x^2 \, y \, dx \]

We can rearrange this to isolate \( dy \):

\[ y \, dx - x^2 \, y \, dx = x \, dy \]

Factoring out \( dx \) on the left side:

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

### Step 2: Dividing Both Sides
Next, we divide both sides by \( y \) and \( x \):

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

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

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

The left side integrates to:

\[ \log |y| + C_1 \]

For the right side, we can split the integral:

\[ \int \frac{1}{x} \, dx - \int x \, dx = \log |x| - \frac{x^2}{2} + C_2 \]

So we have:

\[ \log |y| = \log |x| - \frac{x^2}{2} + C \]

where \( C = C_2 - C_1 \).

### Step 4: Exponentiating Both Sides
To eliminate the logarithm, we exponentiate both sides:

\[ |y| = e^{\log |x| - \frac{x^2}{2} + C} = |x| e^{C} e^{-\frac{x^2}{2}} \]

Let \( k = e^{C} \), we can write:

\[ y = k |x| e^{-\frac{x^2}{2}} \]

### Step 5: General Solution
Thus, the general solution of the differential equation is:

\[ y = k x e^{-\frac{x^2}{2}} \quad \text{(for } x \neq 0\text{)} \]