Applications of Linear Systems with Constant Coefficients 395Substituting for Up and u~ in (13a), u~ = Aup + b , results in
or equivalently(~) f (~ 0 0 ~ 0 ~ -~) 1 (~: e + ~~) ft + (~) 0
h 0 2 0 0 g +ht 0
b = c+ dt
d = -2(g +ht) + 3 = (-2g + 3) - 2htf = g+ ht
h = 2(c+dt) = 2c+2dt.
Equating coefficients in each of these four equations, we find the constants
must simultaneously satisfy(coefficients of t^0 )b = c
d = -2g + 3
f =gh = 2c
and
and
and
and(coefficients of t)O=d
0 = -2h
O=h
0 = 2d.
Solving these eight equations, we find a and e are arbitrary, b = c = d = h = 0,
and f = g = 3/2. Hence,
Up= (e + ~t/2)
3/2is a particular solution of (13a) for any choice of a and e. Choosing a= e = 0,
we obtain the simple particular solution
(15) Up= (3t~2).
3/2The general solution of (13a) is u(t) = uc(t) + up(t), where Uc is given by
equation (14) and where up is given by (15). Imposing the initial conditions
(13b) u(O) = 0 requires c 1 , c 2 , c3, and c 4 to satisfy