——, 
= | x°p(x, x)dxdx 
—c 
E\X*() 
= | Ee + eg(x)} - seta) [pena (12.39) 
From Eq. 12.36 and Eq. 12.37 
x x 
il i |) ae 
PC Grexp -,| Fea ='G exp oe a Ge) 
0 
G(x) = | g(€)de. 
0 
For the linear system, € = 0; therefore from Eq. 12.39, 
loo) 
E[X*(t)] = of - | xeg(x) p(x)dx 
—co 
= ofx-€ E[x(x) a(x]. (12.40) 
For a hard spring system the mean square of the displacement is always smaller than 
it is for the corresponding linear system. 
For the so-called Duffing type system, g{X(t)} = x0), and the equation of motion 
is 
XO) + BX) + w6 {X() + eXx*(0)| = fit). (12.41) 
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