The general solution of the differential
` e^(x)dy + (ye^(x)+2x)dx = 0 ` is

Options

A. `xe^(y)+x^(2)=C`
B. `xe^(y)+y^(2)=C`
C. `ye^(x)+x^(2)=C`
D. `ye^(y)+x^(2)=C`

Explanation

To solve the differential equation \( e^{x} dy + (ye^{x} + 2x)dx = 0 \), we will follow these steps:

1. **Rearranging the Equation**:
Start by rewriting the equation in the standard form:
\[ e^{x} dy + (ye^{x} + 2x)dx = 0 \]
This can be rearranged to:
\[ e^{x} dy = - (ye^{x} + 2x)dx \]
Dividing both sides by \( e^{x} \):
\[ dy = -y dx - \frac{2x}{e^{x}} dx \]

2. **Expressing as a Linear Differential Equation**:
We can express this as:
\[ \frac{dy}{dx} + y = -2x e^{-x} \]
Here, we identify \( P(x) = 1 \) and \( Q(x) = -2x e^{-x} \).

3. **Finding the Integrating Factor**:
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int P(x) dx} = e^{\int 1 dx} = e^{x} \]

4. **Multiplying the Equation by the Integrating Factor**:
Multiply the entire differential equation by the integrating factor:
\[ e^{x} \frac{dy}{dx} + e^{x} y = -2x \]

5. **Recognizing the Left Side as a Product Derivative**:
The left-hand side can be recognized as the derivative of a product:
\[ \frac{d}{dx}(y e^{x}) = -2x \]

6. **Integrating Both Sides**:
Now, integrate both sides with respect to \( x \):
\[ \int \frac{d}{dx}(y e^{x}) dx = \int -2x dx \]
This gives:
\[ y e^{x} = -x^2 + C \]
where \( C \) is the constant of integration.

7. **Solving for \( y \)**:
Finally, we can solve for \( y \):
\[ y = -x^2 e^{-x} + Ce^{-x} \]

Thus, the general solution of the differential equation is:
\[ y e^{x} + x^2 = C \]