8.2.2. 2nd Order LDE

LDEs of the 2nd order are of the form

        y + P 1 ( x ) y + P 2 ( x )y=R( x )

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.

Theorem 8.2.

Consider the DE

        y +a y +by=0                                       (8.3)

where a and b are real constants.  Every solution of (8.3) on ( , )  has the form

        y= e ax/2 [ c 1 u 1 ( x )+ c 2 u 2 ( x ) ]                   (8.4)

where c1 and c2 are constants and the functions u1 and u2 are determined according to the sign of the discriminant d= a 2 4b  as follows:

(a)    If d=0 , then u 1 ( x )=1  and u 2 ( x )=x .

(b)    If d>0 , then u 1 ( x )= e kx  and u 2 ( x )= e kx , where k= 1 2 d .

(c)    If d<0 , then u 1 ( x )=coskx  and u 2 ( x )=sinkx , where k= 1 2 d .

Comment

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

        r 2 +ar+b=0                          (8.5)

with roots

        r 1 = 1 2 ( a+ d )         and        r 2 = 1 2 ( a d )

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 f( x )= e rx , 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

        1 2 ( e ikx ± e ikx )={ coskx isinkx