The integrating factor of the differential equation `(dy)/(dx)+y=(1+y)/(x)`, is

Options

A. `(x)/(e^(x))`
B. `(e^(x))/(x)`
C. `xe^(x)`
D. `e^(x)`

Answer

To find the integrating factor of the differential equation

\[ \frac{dy}{dx} + y = \frac{1+y}{x}, \]

we can follow these steps:

### Step 1: Rewrite the Equation
First, we rewrite the equation in a more standard form. We can separate the terms on the right side:

\[ \frac{dy}{dx} + y = \frac{1}{x} + \frac{y}{x}. \]

### Step 2: Rearranging the Equation
Next, we can rearrange the equation to isolate the \(y\) terms:

\[ \frac{dy}{dx} + y - \frac{y}{x} = \frac{1}{x}. \]

### Step 3: Factor Out \(y\)
Now, we can factor out \(y\) from the left side:

\[ \frac{dy}{dx} + y\left(1 - \frac{1}{x}\right) = \frac{1}{x}. \]

### Step 4: Identify \(p(x)\)
From the standard form of a linear differential equation \(\frac{dy}{dx} + p(x)y = q(x)\), we identify \(p(x)\):

\[ p(x) = 1 - \frac{1}{x}. \]

### Step 5: Find the Integrating Factor
The integrating factor \(I(x)\) is given by:

\[ I(x) = e^{\int p(x) \, dx} = e^{\int \left(1 - \frac{1}{x}\right) \, dx}. \]

### Step 6: Integrate \(p(x)\)
Now we perform the integration:

\[ \int \left(1 - \frac{1}{x}\right) \, dx = \int 1 \, dx - \int \frac{1}{x} \, dx = x - \ln|x| + C. \]

### Step 7: Substitute Back into the Integrating Factor
Now substituting back, we have:

\[ I(x) = e^{x - \ln|x|} = e^x \cdot e^{-\ln|x|} = e^x \cdot \frac{1}{|x|}. \]

Since \(x\) is positive in the context of this problem, we can simplify this to:

\[ I(x) = \frac{e^x}{x}. \]

### Final Answer
Thus, the integrating factor of the differential equation is:

\[ \frac{e^x}{x}. \]

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