Higher Engineering Mathematics, Sixth Edition

448 Higher Engineering Mathematics

Problem 8. (a) The variation of resistance,

Rohms, of an aluminium conductor with

temperatureθ◦Cisgivenby

dR

dθ

=αR,whereα

is the temperature coefficient of resistance of

aluminium. IfR=R 0 whenθ= 0 ◦C, solve the

equation forR.(b)Ifα= 38 × 10 −^4 /◦C, determine

the resistance of an aluminium conductor at 50◦C,

correct to 3 significant figures, when its resistance

at 0◦C is 24.0.

(a)

dR

dθ

=αRis of the form

dy

dx

=f(y)

Rearranging gives: dθ=

dR

αR

Integrating both sides gives:

∫

dθ=

∫

dR

αR

i.e. θ=

1

α

lnR+c,

whichisthegeneral solution.

SubstitutingtheboundaryconditionsR=R 0 when

θ=0gives:

0 =

1

α

lnR 0 +c

from which c=−

1

α

lnR 0

Hence the particular solution is

θ=

1

α

lnR−

1

α

lnR 0 =

1

α

(lnR−lnR 0 )

i.e.θ=

1

α

ln

(

R

R 0

)

orαθ=ln

(

R

R 0

)

Hence eαθ=

R

R 0

from which,R=R 0 eαθ

(b) Substitutingα= 38 × 10 −^4 ,R 0 = 24 .0andθ= 50

intoR=R 0 eαθgives the resistance at 50◦C, i.e.

R 50 = 24 .0e(^38 ×^10

− (^4) × 50 )
= 29 .0ohms
Now try the following exercise
Exercise 178 Further problems on
equations of the form
dy
dx
=f(y)
In Problems 1 to 3, solve the differential
equations.

1. 
   

   dy
   dx
   = 2 + 3 y
   [
   x=
   1
   3
   ln( 2 + 3 y)+c
   ]

   
   


2. 
   

   dy
   dx
   =2cos^2 y [tany= 2 x+c]

   
   


3. 
   

   (y^2 + 2 )

   
   

dy

dx

= 5 y,giveny=1whenx=

1

2

[

y^2

2

+2lny= 5 x− 2

]

4. The current in an electric circuit is given by
   the equation

Ri+L

di

dt

= 0 ,

where L and R are constants. Show that

i=Ie

−Rt

L , given thati=Iwhent=0.

5. The velocity of a chemical reaction is given by
   dx
   dt

=k(a−x),wherexis the amount trans-

ferred in timet,k is a constant anda is

the concentration at timet=0whenx=0.

Solve the equation and determinexin terms

oft.[x=a( 1 −e−kt)]

6. (a) ChargeQcoulombs at timetseconds
   is given by the differential equation
   R

dQ

dt

+

Q

C

=0, whereCis the capaci-

tance in farads andRthe resistance in

ohms. Solve the equation forQgiven

thatQ=Q 0 whent=0.

(b) A circuit possesses a resistance of

250 × 103  and a capacitance of

8. 5 × 10 −^6 F, and after 0.32seconds

the charge falls to 8.0C. Determine

the initial charge and the charge after

1second,each correct to 3 significant

figures.

[(a)Q=Q 0 e

−t

CR(b) 9.30C, 5.81C]

7. A differential equation relating the difference
   in tensionT, pulley contact angleθand coef-
   ficient of frictionμis

dT

dθ

=μT.Whenθ=0,

T=150N, and μ= 0 .30 as slipping starts.

Determine the tension at the point of slipping

whenθ=2radians. Determine also the value

ofθwhenTis 300N. [273.3N, 2.31rads]