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

Options

A. `e^y =e^x + x^3/3+c `
B. `e^y =e^x + 2x + c `
C. `e^y = e^x + x^3 + e `
D. `y=e^x +c`

Answer

To solve the differential equation \(\frac{dy}{dx} = e^{x-y} + x^2 e^{-y}\), we will 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:
\[ e^{x-y} = e^x e^{-y} \]
Thus, the equation becomes:
\[ \frac{dy}{dx} = e^x e^{-y} + x^2 e^{-y} \]

### Step 2: Factor Out Common Terms
Next, we notice that \(e^{-y}\) is a common factor:
\[ \frac{dy}{dx} = e^{-y}(e^x + x^2) \]

### Step 3: Separate Variables
Now, we separate the variables \(y\) and \(x\):
\[ e^y dy = (e^x + x^2) dx \]

### Step 4: Integrate Both Sides
We will now integrate both sides:
\[ \int e^y dy = \int (e^x + x^2) dx \]
The left side integrates to:
\[ e^y + C_1 \]
The right 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 express \(y\) explicitly, 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:
\[ y = \ln\left(e^x + \frac{x^3}{3} + C\right) \]