From u′′+3u′+6u=0 take the coefficients 1, 3 and 6 and use them to write the quadratic equation
1m^2 + 3m + 6 = 0. Solve this quadratic for m:
Using the quadratic formula,
-3 plus or minus sqrt(9-4(1)(6)) -3 plus or minus i*sqrt(15)
m= ------------------------------------------- = -------------------------------------
2 2
-3 sqrt(15)
m = ---- + i ------------- the other root is the same, except that a - sign
2 2 precedes the imaginary part.
This complex root yields a decaying exponential / trig function resembling
(-3/2) (i2) sqrt(15).
e * e
The complex root with the - sign yields a similar decaying exponential / trig function.