The integrating factor of the differential equation `(dy)/(dx)=1/(x+y+2)` is

Options

A. `e^(x+y+2)`
B. `e^y`
C. `e^(-y)`
D. log |x+y+2|

Answer

To find the integrating factor of the differential equation \(\frac{dy}{dx} = \frac{1}{x + y + 2}\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given differential equation:
\[ \frac{dy}{dx} = \frac{1}{x + y + 2} \]
We can rearrange this to express it in a different form:
\[ \frac{dx}{dy} = x + y + 2 \]
This is our first equation.

### Step 2: Identify \(p(y)\) and \(q(y)\)
From the equation \(\frac{dx}{dy} = x + y + 2\), we can identify:
- \(p(y) = -1\) (the coefficient of \(x\))
- \(q(y) = y + 2\)

### Step 3: Write the standard form
The standard form of a first-order linear differential equation is:
\[ \frac{dx}{dy} + p(y)x = q(y) \]
Substituting our values, we have:
\[ \frac{dx}{dy} - x = y + 2 \]

### Step 4: Calculate the integrating factor
The integrating factor \(\mu(y)\) is given by:
\[ \mu(y) = e^{\int p(y) \, dy} \]
Substituting \(p(y) = -1\):
\[ \mu(y) = e^{\int -1 \, dy} = e^{-y} \]

### Step 5: Final answer
Thus, the integrating factor of the differential equation is:
\[ \mu(y) = e^{-y} \]