"Ok Hermes, it's time to stop, if you don't I'll have to do this.

In solving the linear second-order homogeneous differential equation with constant coefficients,

ay″+by′+cy=0, (1)

in the case of complex characteristic roots, we will encounter (nonreal) complex-valued solutions. It turns out, however, that the real and imaginary parts of these objects are also solutions and are, in fact, the actual real solutions that we want. Of course, we can always just verify directly, by substitution, that the real parts of the imaginary parts satisfy (1). But there is a general principle that can be checked too, If u(t)+iv(t) is a solution of (1), then u(t) and v(t) are individual solutions as well, because

a(u+iv)″+b(u+iv)′+c(u+iv)= (au″+bu′+c)+i(av″+bv′+cv),

and a complex number is zero if and only if both its real and imaginary parts are zero.

In the previous section, we studied solutions of equation (1) for the cases in which the discriminant Δ=b²-4ac is positive or zero. We now complete the discussion with Case 3, supposing that Δ<0."