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

Options

A. `(x)/(e^(x))`
B. `(e^(x))/(x)`
C. `x e^(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 in standard form
We start by rewriting the given equation:

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

We can separate the terms on the right side:

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

### Step 2: Combine like terms
Next, we can combine the \(y\) terms:

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

This simplifies to:

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

### Step 3: Identify \(P(x)\) and \(Q(x)\)
From the standard form of a linear first-order differential equation:

\[ \frac{dy}{dx} + P(x)y = Q(x), \]

we can identify:

\[ P(x) = 1 - \frac{1}{x} \quad \text{and} \quad Q(x) = \frac{1}{x}. \]

### Step 4: Find the integrating factor
The integrating factor \(I.F.\) is given by:

\[ I.F. = e^{\int P(x) \, dx}. \]

Substituting \(P(x)\):

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

### Step 5: Calculate the integral
Now we calculate the integral:

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

### Step 6: Substitute back into the integrating factor formula
Now substituting back into the integrating factor formula:

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

Since \(x\) is positive in the context of this problem, we can drop the absolute value:

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

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

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

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