1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Equations with Constant Coefficients 293

Since the set {sinw*t, cosw*t} is a linearly independent set, we equate coeffi-

cients and find

F = E and G = 0.

d - a(w*)^2

So when w* =j:. w, a particular solution of equation (25) is

Yp(t) = d _ ~w*) 2 sinw*t,

and the general solution of equation (25) is

y(t) = Yc(t) + Yp(t) =A sin wt+ B cos wt+ d _ ~w) 2 sinwt

where the constants A and B depend on the initial conditions. Notice that

for w* =j:. w, the solution, y(t), of equation (25) remains bounded for all time,

t.

Case 2. If w* = w- that is, if the frequency of the forcing function is

identical to the natural frequency of the system, then a phenomenon known

as resonance occurs and the particular solution of equation (25) will have

the form

yp(t) = Mtsinw*t + Ntcosw*t.

Differentiating this equation twice, substituting Yp and y~ into equation (25),

and solving, we get M = 0 and N = -E/(2aw). So when w = w the general

solution of equation (25) is

E

y(t) = Asinwt + Bcoswt - - - tcosw*t.

2aw*

Due to the factor tcosw*t, the solution y(t) oscillates with unbounded am-

plitude as t -t oo regardless of the initial conditions which merely determine

the constants A and B.

6.1.2.2 Damped Forced Motion

When the periodic external force f(t) = Esinw*t and when there is a

damping force (b =j:. O),which is the case for all realizable systems, equation (24)
becomes

(26) ay" +by'+ dy = Esinw*t

where a, b, and d are positive constants and E and w* are constants. In the
previous section, we found that the solution of the associated homogeneous
equation, the complementary solution Ye of equation (26), depends on the sign
of b^2 - 4ad.

If b^2 - 4ad < 0, then

Yc(t) = e-"'t(A sin wt+ B cos wt) = Ce-at sin (wt+¢)