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To solve this problem, we need to analyze each option and verify which one correctly represents the solution to the given differential equations.

### Step-by-step Solution:

1. **Option A:**
   The differential equation is \(y' + y = \frac{1}{1 + e^x}\).
   - This is a first-order linear differential equation.
   - The integrating factor \(I(x)\) is \(e^{\int 1 \, dx} = e^x\).
   - Multiply the entire equation by \(e^x\):  
     \[
     e^x y' + e^x y = e^x \cdot \frac{1}{1 + e^x}
     \]
   - The left-hand side is the derivative of \(e^x y\), giving:  
     \[
     \frac{d}{dx}(e^x y) = \frac{e^x}{1 + e^x}
     \]
   - Integrating both sides with respect to \(x\), we use substitution for the right-hand side:  
     Let \(u = 1 + e^x \), \(du = e^x dx\). Then:  
     \[
     \int \frac{e^x}{1 + e^x} \, dx = \int \frac{1}{u} \, du = \ln|u| + C = \ln(1 + e^x) + C
     \]
   - Thus, \(e^x y = \ln(1 + e^x) + C\) and the general solution is:  
     \[
     y(x) = e^{-x} \ln(1 + e^x) + \lambda e^{-x}
     \]
   - This matches the proposed solution in option B, not A.

2. **Option B:**
   The proposed solution for the differential equation \(y' + y =

Which assertion is correct regarding the solution of the differential equations provided?

Learn how to determine the correct solution for a given set of differential equations by analyzing each assertion step-by-step. This problem covers concepts such as differential equations, linear differential equations, and the integrating factor method, suitable for undergraduate or advanced high school students.

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