8.8.3. Case II: Real Roots, Some Repeated

If all roots are real but not distinct, then the uk’s in (8.14) are not independent so that the general solution (8.13) needs to be accordingly modified.

Theorem 8.9.

The m functions

        u k ( x )= x k1 e rx   with k=1,,m

are m independent elements annihilated by ( Dr ) m .

Proof

Independence of the uk’s follows from that of the monomials x k1 .  The proof that each uk is annihilated by ( Dr ) m  is by induction on m.  The case m=1  is trivial.  Thus, assuming case m-1 holds, we wish to prove case m.  To be more specific, we assume the functions u 1 ,, u m1  are all annihilated by ( Dr ) m1 .  Now,

        ( Dr ) u m =( Dr )( x m1 e rx )   =( m1 ) x m2 e rx   =( m1 ) u m1

Hence,

        ( Dr ) m u m =( m1 ) ( Dr ) m1 u m1 =0

where the induction hypothesis was used in the last equality.  QED.

Example 2.

Find the general solution of L( y )=0 , where

        L= D 3 D 2 8D+12

Solution

        L= ( D2 ) 2 ( D+3 )

Hence,

        N( D2 )={ e 2x ,x e 2x }  and      N( D+3 )={ e 3x }

so that the general solution is

        y= c 1 e 2x + c 2 x e 2x + c 3 e 3x

Example 3.

Solve ( D 6 +2 D 5 2 D 3 D 2 )y=0 .

Solution

From

        D 6 +2 D 5 2 D 3 D 2 = D 2 ( D1 ) ( D+1 ) 3

we get

        y= c 1 + c 2 x+ c 3 e x +( c 4 + c 5 x+ c 6 x 2 ) e x