xy′′ + (2x + 1)y ′ + (x + 1)y = 0, y1(x) = e −x .

I'm sorry, but you didn't provide a specific question related to the given differential equation. However, I can tell you that the given equation is a second order linear homogeneous differential equation. The solution to this type of equation is generally in the form of y(x) = C1y1(x) + C2y2(x), where y1(x) and y2(x) are solutions to the differential equation and C1 and C2 are constants.

You've provided one solution, y1(x) = e^-x. To find the second solution, you can use the method of reduction of order, which involves substituting y(x) = y1(x)*v(x) into the differential equation and solving for v(x). Once you have v(x), you can find y2(x) = y1(x)*v(x).

If you need more specific help, please provide more details or a specific question.