The solution of differential equation `(dy)/(dx)=e^(x-y)+x^(2)e^(-y)`is

Options

A. `y=e^(x-y)-x^(2)e^(-y)+C`
B. `e^(y)-e^(x)=(x^(3))/(3)+C`
C. `e^(x)+e^(y)=(x^(3))/(3)+C`
D. `e^(x)-e^(y)=(x^(3))/(3)+C`

Explanation

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

### Step 1: Rewrite the equation
We start with the given equation:
\[ \frac{dy}{dx} = e^{x-y} + x^2 e^{-y} \]
We can rewrite \(e^{x-y}\) as \(e^x e^{-y}\):
\[ \frac{dy}{dx} = e^x e^{-y} + x^2 e^{-y} \]

### Step 2: Factor out \(e^{-y}\)
Next, we factor \(e^{-y}\) out of the right-hand side:
\[ \frac{dy}{dx} = e^{-y}(e^x + x^2) \]

### Step 3: Separate the variables
Now 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 side integrates to:
\[ e^y = \int e^x dx + \int x^2 dx \]
Calculating the right side:
\[ e^y = e^x + \frac{x^3}{3} + C \]

### Step 5: Solve for \(y\)
To express \(y\) in terms of \(x\), we take the natural logarithm of both sides:
\[ y = \ln\left(e^x + \frac{x^3}{3} + C\right) \]

### Final Form
The general solution of the differential equation is:
\[ e^y - e^x = \frac{x^3}{3} + C \]