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

Options

A. `e^(x)= (y^(3))/(3) + e^(y) +c`
B. `e^(x)= (x^(2))/(3) + e^(x) +c`
C. `e^(-y) =- (x^(3))/(3) - e^(x)-c`
D. None of these

Answer

To solve the differential equation \(\frac{dy}{dx} = e^{x+y} + x^2 e^y\), we will follow a series of steps to separate the variables and integrate.

### Step 1: Rewrite the equation
We start with the given equation:
\[ \frac{dy}{dx} = e^{x+y} + x^2 e^y \]
Using the property of exponents, 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 can factor \(e^y\) out of the right-hand side:
\[ \frac{dy}{dx} = e^y (e^x + x^2) \]

### Step 3: Separate variables
Now, we separate the variables \(y\) and \(x\):
\[ \frac{1}{e^y} dy = (e^x + x^2) dx \]

### Step 4: Integrate both sides
We will now integrate both sides:
\[ \int \frac{1}{e^y} dy = \int (e^x + x^2) dx \]
The left side becomes:
\[ \int e^{-y} dy = -e^{-y} + C_1 \]
The right side can be integrated as follows:
\[ \int e^x dx + \int x^2 dx = e^x + \frac{x^3}{3} + C_2 \]
Thus, we have:
\[ -e^{-y} = e^x + \frac{x^3}{3} + C \]
where \(C = C_2 - C_1\).

### Step 5: Solve for \(y\)
Now, we solve for \(y\):
\[ -e^{-y} = e^x + \frac{x^3}{3} + C \]
Multiplying through by \(-1\):
\[ e^{-y} = -\left(e^x + \frac{x^3}{3} + C\right) \]
Taking the natural logarithm:
\[ -y = \ln\left(-\left(e^x + \frac{x^3}{3} + C\right)\right) \]
Thus, we have:
\[ y = -\ln\left(-\left(e^x + \frac{x^3}{3} + C\right)\right) \]

### Final Solution
The solution of the differential equation is:
\[ y = -\ln\left(-\left(e^x + \frac{x^3}{3} + C\right)\right) \]