082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

19.6 Constant coefficient equations 583
EXERCISES19.6.3
1 Find the general solution ofthe followingequations:
(a) d2 x
dt 2 −2dx dt −3x=6
(b) d2 y
dx 2 +5dy dx +4y=8
(c) d2 y
dt 2 +5dy dt +6y=2t
(d) d2 x
dt 2 +11dx dt +30x=8t
(e) d2 y
dx 2 +2dy dx +3y=2sin2x
d 2 y
(f)
dt 2 + dy
dt +y=4cos3t
(g) d2 y
dx 2 +9y=4e8x
(h) d2 x
dt 2 −16x=9e6t
2 Find aparticular integral forthe equation
d 2 x
dt 2 −3dx dt +2x=5e3t
3 Find aparticular integral forthe equation
d 2 x
dt 2 −x=4e−2t
4 Obtainthe general solution of
y ′′ −y ′ −2y=6.
5 Obtainthe general solution ofthe equation
d 2 y
dx 2 +3dy dx +2y=10cos2x
Find the particularsolution satisfying
y(0) =1, dy (0) =0.
dx
6 Find aparticularintegralforthe equation
d 2 y
dx 2 + dy
dx +y=1+x
7 Find the general solution of
d 2 x
(a)
dt 2 −6dx dt +5x=3
d 2 x
(b)
dt 2 −2dx dt +x=et
8 Forthe circuitshown in Figure19.15showthat
RCL d2 i 2
dt 2 +Ldi 2
+Ri
dt 2 =E(t)
IfL=1mH,R=10,C=1Fand
E(t) = 2sin100πt,findthe complementary function.
9 Find the general solution of
i
d 2 i
dt 2 +8di +25i = 48cos3t −16sin3t
dt
R
Figure19.15
i 1
i 2
E(t)
C
L
Solutions
1 (a) x=Ae −t +Be 3t −2
(f) y = e −0.5t (Acos0.866t +Bsin0.866t) −
0.438cos3t +0.164sin3t
(b) y=Ae −x +Be −4x +2
(c) y=Ae −2t +Be −3t + t 3 − 5 (g) y =Acos3x +Bsin3x +0.0548e 8x
18
(d) x =Ae −6t +Be −5t +0.267t −0.0978 (h) x=Ae 4t +Be −4t + 9
20 e6t
(e) y = e −x [Asin √ 2x+Bcos √ 2x] − 8 17 cos2x − 2 x=2.5e 3t
2
17 sin2x 3 x = 4 3 e−2t