Answer

Let's check the initial condition first:

y(0) = 4e⁰ = 4,

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

and

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.