460 Higher Engineering Mathematics
- In an alternating current circuit containing
resistance R and inductanceLthe current
i is given by:Ri+L
di
dt
=E 0 sinωt.Given
i=0whent=0, show that the solution of the
equation is given by:
i=
(
E 0
R^2 +ω^2 L^2
)
(Rsinωt−ωLcosωt)
+
(
E 0 ωL
R^2 +ω^2 L^2
)
e−Rt/L
- The concentration,C, of impurities of an oil
purifier varies with timetand is described by
the equation
a
dC
dt
=b+dm−Cm,wherea,b,dandmare
constants. GivenC=c 0 whent=0, solve the
equation and show that:
C=
(
b
m
+d
)
( 1 −e−mt/a)+c 0 e−mt/a
- The equation of motion of a train is given
by:m
dv
dt
=mk( 1 −e−t)−mcv,wherevis the
speed,tis the time andm,kandcare con-
stants. Determine the speed,v,givenv=0at
t=0. [
v=k
{
1
c
−
e−t
c− 1
+
e−ct
c(c− 1 )
}]