Integrating factor of differential equation :
`(dy)/(dx)+y=3` is :

Options

A. `x`
B. e
C. `e^(x)`
D. `log x`.

Answer

To find the integrating factor of the differential equation \(\frac{dy}{dx} + y = 3\), we will follow these steps:

### Step 1: Identify the standard form
The given differential equation is:
\[ \frac{dy}{dx} + y = 3 \]
This can be compared to the standard form of a linear first-order differential equation:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
where \(P(x) = 1\) and \(Q(x) = 3\).

### Step 2: Determine the integrating factor
The integrating factor \(I(x)\) is given by the formula:
\[ I(x) = e^{\int P(x) \, dx} \]
Substituting \(P(x) = 1\):
\[ I(x) = e^{\int 1 \, dx} = e^{x} \]

### Step 3: Write the final answer
Thus, the integrating factor of the differential equation \(\frac{dy}{dx} + y = 3\) is:
\[ e^{x} \]