LDEs of the 2nd order are of the form
If the coefficients P1, P2 and R are continuous over some open interval J, then an existence theorem guarantees that solutions always exists over J [see §8.5]. However, there is no general formula analogous to (8.2) so that the theory is far from complete, except in special cases.
Consider the DE
where a and b are real constants. Every solution of (8.3) on has the form
where c1 and c2 are constants and the functions u1 and u2 are determined according to the sign of the discriminant as follows:
(a) If , then and .
(b) If , then and , where .
(c) If , then and , where .
By replacing each nth order derivative y(n) by the nth power monomial rn, we turn a LDE with constant coefficients into a polynomial equation, known as the characteristic equation of the LDE. For eq(8.3), this is
Thus, d is so-called because it is actually the discriminant of (8.5) that determines the nature of its solutions.
Assuming we are working with complex numbers, it is then easily shown that all solutions of a LDE with constant coefficients are linear combinations of functions of the form , where r is a root of the characteristic equation of the LDE. Theorem 8.2 then follows by re-expressing everything in terms of real functions using