Higher Engineering Mathematics

LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS 459

I

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)

}]