Assertion: The general solution of `( dy ) /( dx) + y =1 `is ` y e^x = e ^(X)+C `
Reason: The number of arbitrary constant in the general solution of the differential equation is equal to the order of D.E.

Answer

To solve the differential equation \( \frac{dy}{dx} + y = 1 \), we will follow these steps:

### Step 1: Identify the form of the equation
The given equation is a first-order linear ordinary differential equation. It can be written in the standard form:
\[ \frac{dy}{dx} + Py = Q \]
where \( P = 1 \) and \( Q = 1 \).

**Hint:** Recognize the standard form of a first-order linear differential equation.

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

**Hint:** The integrating factor is crucial for solving linear differential equations.

### Step 3: Multiply the entire equation by the integrating factor
Multiply the original equation by \( e^{x} \):
\[ e^{x} \frac{dy}{dx} + e^{x} y = e^{x} \]

**Hint:** This step transforms the left side into a derivative of a product.

### Step 4: Rewrite the left side as a derivative
The left side can be rewritten as:
\[ \frac{d}{dx}(y e^{x}) = e^{x} \]

**Hint:** Recognizing the left side as a derivative simplifies the integration process.

### Step 5: Integrate both sides
Integrate both sides with respect to \( x \):
\[ \int \frac{d}{dx}(y e^{x}) \, dx = \int e^{x} \, dx \]
This gives:
\[ y e^{x} = e^{x} + C \]
where \( C \) is the constant of integration.

**Hint:** Remember to include the constant of integration when you integrate.

### Step 6: Solve for \( y \)
Now, we can solve for \( y \):
\[ y = \frac{e^{x} + C}{e^{x}} = 1 + Ce^{-x} \]

**Hint:** Rearranging the equation helps isolate the dependent variable.

### Conclusion
The general solution of the differential equation \( \frac{dy}{dx} + y = 1 \) is:
\[ y = 1 + Ce^{-x} \]

### Assertion and Reason
- **Assertion:** The general solution of \( \frac{dy}{dx} + y = 1 \) is \( y e^{x} = e^{x} + C \) is correct.
- **Reason:** The number of arbitrary constants in the general solution of the differential equation is equal to the order of the D.E. is also true, but it does not explain the assertion directly.

### Final Statement
Both the assertion and the reason are correct, but the reason does not provide a direct explanation for the assertion.