12. $y^{\prime \prime}+3 y^{\prime}=0, y(0)=-2, y^{\prime}(0)=3$
Find roots of characteristic equation:
$$
\begin{array}{l}{r^{2}+3 r=0} \\ {r(r+3)=0} \\ {r_{1}=0} \\ {r_{2}=-3}\end{array}
$$
General solution of equation is:
$$
y=c_{1}+c_{2} e^{-3 t}
$$
Initial terms:
$$
\begin{array}{l}{y(0)=-2} \\ {-2=c_{1}+c_{2} e^{0}} \\ {-2=c_{1}+c_{2}}\end{array}
$$
$$
\begin{array}{l}{y^{\prime}(0)=3} \\ {-3 c_{2} e^{-3 t}=3} \\ {-3 c_{2} e^{0}=3} \\ {-3 c_{2}=3}\end{array}
$$
$$
\begin{array}{l}{c_{2}=-1} \\ {c_{1}+c_{2}=-2} \\ {c_{1}=-2+1=-1}\end{array}
$$
Solution of problem: $$y=-1-e^{-3 t}$$
