Higher Engineering Mathematics, Sixth Edition

460 Higher Engineering Mathematics

7. 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

8. 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

9. 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 )

}]