2.10. For each homogeneous differential equation given below, find the characteristic equation and show that at least some of its roots are complex. Find the homogeneous solution for $t \ge 0$ in each case subject to the initial conditions specified.
a. $\frac{d^2y(t)}{dt^2} + 3y(t) = 0$, $y(0) = 2$, $\frac{dy(t)}{dt}|_{t=0} = 0$
b. $\frac{d^2y(t)}{dt^2} + 2\frac{dy(t)}{dt} + 2y(t) = 0$, $y(0) = -2$, $\frac{dy(t)}{dt}|_{t=0} = -1$