### 8.8.4. Case III: Complex Roots

If complex exponentials are used, there is no need to distinguish between real and complex roots.  If real-valued solutions are desired, each pair of conjugate roots α±iβ  should be combined into a quadratic factor

Q= D 2 2αD+ α 2 + β 2                  (8.15)

whose null space is N( Q )={ e αx cosβx  ,   e αx sinβx } .  In case of multiplicity m, a simple generalization of Theorem 8.9 gives

N( Q m )={ x k1 e αx cosβx  ,   x k1 e αx sinβx  ;  k=1,,m }

#### Example 4.

y 4 y +13 y =0

The associated polynomial is r 3 4 r 2 +13r , with roots 0, and 2±3i .  The general solution is

y= c 1 + e 2x ( c 2 cos3x+ c 3 sin3x )

#### Example 5.

y 2 y +4 y 8y=0

The associated polynomial is r 3 2 r 2 +4r8=( r2 )( r 2 +4 ) , with roots 2, and ±2i .  The general solution is

y= c 1 e 2x + c 2 cos2x+ c 3 sin2x

#### Example 6.

y ( 5 ) 9 y ( 4 ) +34 y 66 y +65 y 25y=0

The associated polynomial is ( r1 ) ( r 2 4r+5 ) 2 , with roots 1, 2±i  and 2±i .  The general solution is

y= c 1 e x + e 2x [ ( c 2 + c 3 x )cosx+( c 4 + c 5 x )sinx ]