Determine if the function *y* = 4*e*^{ - x} is a solution to the IVP

*y*" + *y* ' = 0 and *y*(0) = 4.

Answer

Let's check the initial condition first:

*y*(0) = 4*e*^{0} = 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*' = -4*e*^{-x}

and

*y*" = 4*e*^{-x},

so the left-hand side of the d.e. is

*y*" + *y*' = 4*e*^{-x} + (-4*e*^{-x}) = 0.

Since the left- and right-hand sides of the d.e. come out the same, the function *y* = 4*e*^{-x} satisfies the differential equation. Since the function *y* = 4*e*^{-x} satisfies both the d.e. and the initial condition, this function is a solution to the IVP.