
582 Appendix 1
We now distinguish between the cases in which the roots are real and distinct, complex, or
coincident.
(i) The Case of Real and Distinct Roots (a
2
– 4b > 0)
In this case, we have two independent solutions y
1
= exp(p
1
t) and y
2
= exp(p
2
t), and the
general solution of Equation A1.16 is a linear combination of these two:
y = A exp(p
1
t) + B exp(p
2
t) (A1.55)
where A and B are constants.
(ii) The Case of Complex roots (a
2
– 4b < 0)
If the roots p
1
and p
2
of the auxiliary equation are imaginary, the solution given by
Equation A1.54 is still correct. In order to give the solutions in terms of real quantities,
we can use the Euler relationships ...