Solution of `x(dy)/(dx)+y=xe^(x)`, is

Options

A. `xy=e^(x)(x+1)+C`
B. `xy=e^(x)(x-1)+C`
C. `xy=e^(x)(1-x)+C`
D. `xy=e^(y)(y-1)+C`

Answer

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

### Step 1: Rewrite the Equation
First, we can rewrite the given equation in a more standard form. We divide the entire equation by \( x \) (assuming \( x \neq 0 \)):
\[ \frac{dy}{dx} + \frac{y}{x} = e^x \]

### Step 2: Identify \( p(x) \) and \( q(x) \)
Now, we can identify the functions \( p(x) \) and \( q(x) \) from the standard form of a linear differential equation:
\[ \frac{dy}{dx} + p(x) y = q(x) \]
Here, we have:
- \( p(x) = \frac{1}{x} \)
- \( q(x) = e^x \)

### Step 3: Find the Integrating Factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int \frac{1}{x} \, dx} = e^{\log |x|} = |x| \]
Since we are assuming \( x > 0 \), we can simplify this to:
\[ \mu(x) = x \]

### Step 4: Multiply the Equation by the Integrating Factor
Now, we multiply the entire differential equation by the integrating factor \( x \):
\[ x \frac{dy}{dx} + y = x e^x \]

### Step 5: Rewrite the Left Side
The left side can be rewritten as the derivative of a product:
\[ \frac{d}{dx}(xy) = x e^x \]

### Step 6: Integrate Both Sides
Now, we integrate both sides with respect to \( x \):
\[ \int \frac{d}{dx}(xy) \, dx = \int x e^x \, dx \]
The left side simplifies to:
\[ xy = \int x e^x \, dx \]
To solve the right side, we can use integration by parts. Let:
- \( u = x \) and \( dv = e^x \, dx \)
Then:
- \( du = dx \) and \( v = e^x \)

Using integration by parts:
\[ \int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C \]
Thus:
\[ xy = x e^x - e^x + C \]

### Step 7: Solve for \( y \)
Now, we solve for \( y \):
\[ y = e^x - \frac{e^x}{x} + \frac{C}{x} \]

### Final Solution
The solution to the differential equation \( x \frac{dy}{dx} + y = x e^x \) is:
\[ y = e^x \left( x - 1 \right) + \frac{C}{x} \]