294 Ordinary Differential Equations
where a= b/(2a), w = v'4ad-b^2 /(2a), and A , B, C , and</> are arbitrary
constant s.
If b^2 - 4ad = 0, then
Yc(t) = (A+ Bt)e-at
where a= b/(2a) and A and Bare arbitrary constants.
If b^2 - 4ad > 0, then
Yc(t) = Aerit + Ber2twhere r 1 = (-b + v'b^2 - 4ad)/(2a) < 0, r 2 = (-b -v'b^2 - 4ad)/(2a) < 0, and
A and B are arbitrary constants.
Notice that as t -+ oo, Yc(t) -+ 0 regardless of the value of the quantity
b^2 - 4ad and regardless of t he form of the complementary solution Yc(t).
Because of this property the term Yc(t) of the general solution y(t) = Yc(t) +
yp(t) is call ed t he transient solution- the function Yc(t) and its effects die
out with increasing time.
There is a particular solution of equation (26) of t he formyp(t) = F sinwt + G coswt.
Differentiating twice; substituting yp, y~, and y~ into equation (26); equating
coefficients of sinwt and coswt; and solving for F and G it can be shown
that
F = (d - a(w)^2 )E
H(w)
-bw*E
and G = H(w*)where H(w) = [d - a(w)^2 ]^2 + b^2 (w*)^2. So a particular solut ion of equation
(26) isyp(t) = E[(d-a(w)^2 ) sinwt - bw coswt]/ H(w*)
or equivalently
(27) Yp(t) = [Esin (wt + </>)]/JH(w)
where </> simultaneously satisfies
cos</>= (d - a(w)^2 )/ J H(w) and sin</>= -bw / JH(w).
The general solution of equation (26) is y(t) = Yc(t) + yp(t). The transient
solution (complementary solution, Ye) contains arbitrary constants A and B
which depend upon the initial condit ions under which the system was started.
Since the initial conditions affect only the transient solution and since the
transient solution approaches zero after a sufficiently long period of time, the
init ial conditions influence the solution of equation (26) only for a "short"
period of time. For all practical purposes after a sufficiently long period of
time the general solution becomes the particular solution. For this reason, the
particular solution is called the steady state solution. Consequently, after