460 Higher Engineering Mathematics
7. In an alternating current circuit containing
resistance R and inductance L the current
i is given by: Ri + L
di
dt
= E 0 sin ωt . Given
i = 0 when t = 0, show that the solution of the
equation is given by:
i =
E 0
R 2 + ω 2 L 2
(R sin ωt − ωL cosωt )
+
E 0 ωL
R 2 + ω 2 L 2
e
− Rt/L
8. The concentration, C, of impurities of an oil
purifier varies with time t and is described by
the equation
a
dC
dt
= b + dm − Cm, where a, b, d and m are
constants. Given C = c 0 when t = 0, solve the
equation and show that:
C =
b
m
+ d
(1 − e
−mt/a
) + c 0 e
−mt/a
9. The equation of motion of a train is given
by: m
dv
dt
= mk(1 − e −t ) − mcv, where v is the
speed, t is the time and m, k and c are constants. Determine the speed, v, given v = 0 at
t = 0.
v = k
1
c
−
e −t
c − 1
+
e −ct
c(c − 1)
7. In an alternating current circuit containing
resistance R and inductance L the current
i is given by: Ri + L
di
dt
= E 0 sin ωt . Given
i = 0 when t = 0, show that the solution of the
equation is given by:
i =
E 0
R 2 + ω 2 L 2
(R sin ωt − ωL cosωt )
+
E 0 ωL
R 2 + ω 2 L 2
e
− Rt/L
8. The concentration, C, of impurities of an oil
purifier varies with time t and is described by
the equation
a
dC
dt
= b + dm − Cm, where a, b, d and m are
constants. Given C = c 0 when t = 0, solve the
equation and show that:
C =
b
m
+ d
(1 − e
−mt/a
) + c 0 e
−mt/a
9. The equation of motion of a train is given
by: m
dv
dt
= mk(1 − e −t ) − mcv, where v is the
speed, t is the time and m, k and c are constants. Determine the speed, v, given v = 0 at
t = 0.
v = k
1
c
−
e −t
c − 1
+
e −ct
c(c − 1)