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

Options

A. `y=(e^x)/(x)+(k)/(x)`
B. `y=xe^x+Cx`
C. `y=xe^x+k`
D. `x=(e^y)/(y)+k/y`

Answer

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

### Step 1: Rewrite the equation
We start with the given equation:
\[ x \frac{dy}{dx} + y = e^x \]
To simplify, we divide the entire equation by \( x \):
\[ \frac{dy}{dx} + \frac{y}{x} = \frac{e^x}{x} \]

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

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

### Step 4: Multiply the equation by the integrating factor
We multiply the entire equation by the integrating factor \( x \):
\[ x \frac{dy}{dx} + y = e^x \]

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

### Step 6: Integrate both sides
Now we integrate both sides with respect to \( x \):
\[ \int \frac{d}{dx}(xy) \, dx = \int e^x \, dx \]
This gives us:
\[ xy = e^x + C \]
where \( C \) is the constant of integration.

### Step 7: Solve for \( y \)
To find \( y \), we divide both sides by \( x \):
\[ y = \frac{e^x}{x} + \frac{C}{x} \]
We can rewrite \( C \) as \( k \) (where \( k \) is another constant):
\[ y = \frac{e^x}{x} + \frac{k}{x} \]

### Final Solution
Thus, the solution of the differential equation is:
\[ y = \frac{e^x}{x} + \frac{k}{x} \]