Solve the following differential equations :
`xdy+(y-x^(2)y)dx=0`.

Answer

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

### Step 1: Rearrange the equation
Start by rewriting the equation in a more manageable form:
\[ x \, dy = (y - x^2 y) \, dx \]
This can be simplified to:
\[ x \, dy = y(1 - x^2) \, dx \]

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

### Step 3: Integrate both sides
Next, we integrate both sides:
\[ \int \frac{dy}{y} = \int \frac{(1 - x^2)}{x} \, dx \]
The left side integrates to:
\[ \log |y| + C_1 \]
The right side can be split into two parts:
\[ \int \frac{1}{x} \, dx - \int x \, dx = \log |x| - \frac{x^2}{2} + C_2 \]
Combining the constants \( C_1 \) and \( C_2 \) into a single constant \( C \), we have:
\[ \log |y| = \log |x| - \frac{x^2}{2} + C \]

### Step 4: Exponentiate to solve for \( y \)
Now, we exponentiate both sides to solve for \( y \):
\[ |y| = e^{\log |x| - \frac{x^2}{2} + C} = |x| e^{C} e^{-\frac{x^2}{2}} \]
Let \( k = e^{C} \), then:
\[ y = k |x| e^{-\frac{x^2}{2}} \]

### Step 5: Final solution
Thus, the general solution to the differential equation is:
\[ y = C x e^{-\frac{x^2}{2}} \]
where \( C \) is a constant.