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

Options

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

Explanation

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

### Step 1: Rearrange the Equation
We start with the given equation:
\[ x \frac{dy}{dx} + y = e^x \]
We can rearrange this to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} + \frac{y}{x} = \frac{e^x}{x} \]

### Step 2: Identify the Integrating Factor
This is a linear first-order differential equation of the form \( \frac{dy}{dx} + P(x)y = Q(x) \), where \( P(x) = \frac{1}{x} \) and \( Q(x) = \frac{e^x}{x} \).

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 take \( \mu(x) = x \).

### Step 3: Multiply the Equation by the Integrating Factor
Now, we multiply the entire equation by the integrating factor \( x \):
\[ x \frac{dy}{dx} + y = e^x \]

### Step 4: Rewrite the Left Side as a Derivative
The left side can be rewritten as the derivative of a product:
\[ \frac{d}{dx}(xy) = e^x \]

### Step 5: Integrate Both Sides
Now we integrate both sides:
\[ \int \frac{d}{dx}(xy) \, dx = \int e^x \, dx \]
This gives us:
\[ xy = e^x + C \]
where \( C \) is the constant of integration.

### Step 6: Solve for \( y \)
To express \( y \) explicitly, we divide both sides by \( x \):
\[ y = \frac{e^x}{x} + \frac{C}{x} \]

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

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