Question

Write the given second order equation as its equivalent system of first order equations. u′′+3u′+6u=0

Answer 1

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.