Step-by-step Solution:

Given differential equation: \[ (e^x + 1), y, dy = (y + 1), e^x , dx \] Step 1: Separate variables Bring all (y)-terms to one side and (x)-terms to the other: \[ \frac{y}{y+1}, dy = \frac{e^x}{e^x+1}, dx \] Step 2: Integrate both sides Left side \[ \frac{y}{y+1} = 1 - \frac{1}{y+1} \] \[ \int \left(1 - \frac{1}{y+1}\right) dy = y - \ln(y+1) \] Right side \[ \int \frac{e^x}{e^x+1} dx = \ln(e^x+1) \] Step 3: Combine results \[ y - \ln(y+1) = \ln(e^x+1) + C \] Rewriting: \[ \ln(y+1) + \ln(e^x+1) = y + C \] \[ \ln\big((y+1)(e^x+1)\big) = y + C \] Exponentiating: \[ (y+1)(e^x+1) = C e^{y} \] or \[ {e^{y} = C (e^x+1)(y+1)} \]