One of the following is a general solution of the homogeneous differential equation $y''+3y'+2y=0$
$$\begin{aligned} y&=ae^{x}+be^{1x} \\ y&=ae^{-2x}+be^{-1x} \\ y&=ae^{x}+be^{-1x} \\ y&=ae^{-2x}+be^{1x} \\ \end{aligned}$$ and one of the following is a solution to the nonhomogeneous equation $y''+3y'+2y=-12e^{x}$ $$\begin{aligned} y&=-2e^{x} \\ y&=-6e^{x} \\ y&=-8e^{x} \\ y&=-4e^{x} \\ \end{aligned}$$ By superposition, the general solution of the equation $y''+3y'+2y=-12e^{x}$ is
$y =$
Find the solution with
$$\begin{aligned} y(0)&=7 \\ y^\prime(0)&=8 \\ \end{aligned}$$ $y =$

Being the highly trained differential equations fighting machine that I am, I checked the Wronskian (using the solutions to the homogeneous equation without the coefficients a and b) for good measure, and found it to be . The fundamental theorem tells me that that this is the unique solution to the IVP on the interval .