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 can follow these steps:

### Step 1: Rewrite the equation
We start with the given differential equation:
\[ x \frac{dy}{dx} + y = x e^x \]

### Step 2: Divide by \( x \)
To simplify the equation, we divide every term by \( x \):
\[ \frac{dy}{dx} + \frac{y}{x} = e^x \]

### Step 3: Identify \( p(x) \) and \( q(x) \)
Now, we can identify \( p(x) \) and \( q(x) \) from the standard form of a linear differential equation:
\[ \frac{dy}{dx} + p(x)y = q(x) \]
Here, \( p(x) = \frac{1}{x} \) and \( q(x) = e^x \).

### Step 4: Calculate the integrating factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int \frac{1}{x} \, dx} = e^{\ln |x|} = |x| \]
Since \( x > 0 \), we can take \( \mu(x) = x \).

### Step 5: Multiply the entire equation by the integrating factor
Now we multiply the entire equation by the integrating factor \( x \):
\[ x \frac{dy}{dx} + y = x e^x \]

### Step 6: Rewrite the left side
The left-hand side can be rewritten as:
\[ \frac{d}{dx}(xy) = x e^x \]

### Step 7: Integrate both sides
Now, we integrate both sides:
\[ \int \frac{d}{dx}(xy) \, dx = \int x e^x \, dx \]
The left side simplifies to \( xy \). For the right side, we will use integration by parts:
Let \( u = x \) and \( dv = e^x dx \), then \( du = dx \) and \( v = e^x \).
Using integration by parts:
\[ \int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C \]

### Step 8: Write the general solution
Thus, we have:
\[ xy = x e^x - e^x + C \]
Dividing through by \( x \) gives:
\[ y = e^x - \frac{e^x}{x} + \frac{C}{x} \]

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