Consider the following linear second-order homogeneous differential equation with constant coefficients and two initial conditions

$\frac{{d}^{2}y\left(t\right)}{{dt}^{2}}-\frac{1}{3}\cdot \frac{dy\left(t\right)}{dt}-\frac{2}{9}\cdot y\left(t\right)=0,\text{}y\left(0\right)=0,\text{}\frac{dy\left(0\right)}{dt}=-3$