Find the integrating factor of the differential equation :
`y(dx)/(dy)-2x=y^(3) e^(-y)`.

Answer

To find the integrating factor of the differential equation given by:

\[ y \frac{dx}{dy} - 2x = y^3 e^{-y} \]

we will follow these steps:

### Step 1: Rewrite the equation in standard form
First, we will divide the entire equation by \( y \) to simplify it:

\[ \frac{dx}{dy} - \frac{2x}{y} = y^2 e^{-y} \]

### Step 2: Identify \( p(y) \) and \( q(y) \)
Now, we can identify \( p(y) \) and \( q(y) \) from the standard form of a linear first-order differential equation:

\[ \frac{dx}{dy} + p(y)x = q(y) \]

Here, we have:

- \( p(y) = -\frac{2}{y} \)
- \( q(y) = y^2 e^{-y} \)

### Step 3: Calculate the integrating factor
The integrating factor \( \mu(y) \) is given by the formula:

\[ \mu(y) = e^{\int p(y) \, dy} \]

Substituting \( p(y) \):

\[ \mu(y) = e^{\int -\frac{2}{y} \, dy} \]

### Step 4: Evaluate the integral
Now, we evaluate the integral:

\[ \int -\frac{2}{y} \, dy = -2 \ln |y| = \ln |y|^{-2} \]

### Step 5: Substitute back into the integrating factor formula
Now substituting back into the formula for the integrating factor:

\[ \mu(y) = e^{\ln |y|^{-2}} = |y|^{-2} \]

Since \( y \) is positive in our context, we can drop the absolute value:

\[ \mu(y) = \frac{1}{y^2} \]

### Final Result
Thus, the integrating factor of the differential equation is:

\[ \mu(y) = \frac{1}{y^2} \]

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