Answer

If y = e^x then y ' = e^x and y" = e^x. The left-hand side of the d.e. comes out to be

y' + y" = e^x + e^x = 2e^x

and the right-hand side of the d.e. comes out to be

2y = 2(e^x) = 2e^x.

Since the left-hand side and right-hand side of the d.e. came out the same, y = e^x is a solution to this differential equation.