so the initial condition is satisfied. Now for the differential equation. The right-hand side of the differential equation is 0. We have
y' = -4e-x
y" = 4e-x,
so the left-hand side of the d.e. is
y" + y' = 4e-x + (-4e-x) = 0.
Since the left- and right-hand sides of the d.e. come out the same, the function y = 4e-x satisfies the differential equation. Since the function y = 4e-x satisfies both the d.e. and the initial condition, this function is a solution to the IVP.