The integrating factor of `(dy)/(dx) + y = e ^(-x) is `

Options

A. `x `
B. `-x`
C. `e ^(x)`
D. `y = e ^(-x)`

Answer

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

### Step 1: Identify \(p(x)\)
The given equation can be rewritten in the standard form of a linear differential equation:
\[ \frac{dy}{dx} + p(x)y = q(x) \]
Here, we can identify \(p(x) = 1\) and \(q(x) = e^{-x}\).

### Step 2: Calculate the Integrating Factor
The integrating factor \(\mu(x)\) is given by the formula:
\[ \mu(x) = e^{\int p(x) \, dx} \]
In our case, since \(p(x) = 1\), we have:
\[ \mu(x) = e^{\int 1 \, dx} \]

### Step 3: Perform the Integration
Now we perform the integration:
\[ \int 1 \, dx = x \]
Thus, the integrating factor becomes:
\[ \mu(x) = e^{x} \]

### Step 4: Write the Final Result
The integrating factor for the differential equation \(\frac{dy}{dx} + y = e^{-x}\) is:
\[ \mu(x) = e^{x} \]