Solve the following differential equations.
`(dy)/(dx )=e^(x-y)+x^2e^(-y)`

Answer

To solve the differential equation

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

we can start by rewriting the equation in a more manageable form.

### Step 1: Rewrite the equation
We can express \( e^{x-y} \) as \( e^x e^{-y} \). Therefore, we can rewrite the equation as:

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

### Step 2: Factor out \( e^{-y} \)
Now, we can factor out \( e^{-y} \) from the right-hand side:

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

### Step 3: Separate variables
Next, we can separate the variables by multiplying both sides by \( e^y \) and \( dx \):

\[ e^y dy = (e^x + x^2) dx \]

### Step 4: Integrate both sides
Now, we integrate both sides:

\[ \int e^y dy = \int (e^x + x^2) dx \]

The left-hand side integrates to:

\[ e^y + C_1 \]

The right-hand side integrates to:

\[ e^x + \frac{x^3}{3} + C_2 \]

Combining the constants of integration, we have:

\[ e^y = e^x + \frac{x^3}{3} + C \]

### Step 5: Solve for \( y \)
To find \( y \), we take the natural logarithm of both sides:

\[ y = \ln\left(e^x + \frac{x^3}{3} + C\right) \]

### Final Solution
Thus, the solution to the differential equation is:

\[ y = \ln\left(e^x + \frac{x^3}{3} + C\right) \]