Of _ Oe thi) | itd: wots 
WiAji—-A3) i-me)? | 1-we 1 -pye 
(6.82) 
Gr Ge) | ae Phe 
2A2Aj—AS) 1-2)” | 1-n3 1- mye 
Taking the ratio on both sides of the two equations in Eq. 6.82 gives 
ry Mtb) woth 
bi) et 2 re eare 
GAs 2 NE ore ae (6.83) 
a b b 
+b Hot0; Mith 
(U2 + by) rae ei 
This is a quadratic equation in b, and has the form 
b3+2Pb,+1=0. (6.84) 
Thus 2P =-| by Mes s (6.85) 
b; 
After manipulation, we get 
op = Ai +e =H») SG LIES) (6.86) 
Ay — 13) —Anu2(1 — 7) 
and also 
bj =-P + VP*-1. (6.87) 
The homogeneous equations of Eqs. 6.53 or 6.54 is 
(D? + aD + ao)X(t) = 0 
(D? + 2k, + w2)X(t) = 0, 
and its eigenvalues A;,A2 are, according to Eq. 6.65, 
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