Higher Engineering Mathematics

SOLUTION OF FIRST ORDER DIFFERENTIAL EQUATIONS BY SEPARATION OF VARIABLES 447

I

αis the temperature coefficient of resistance of

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

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

mine 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,

which is the general solution.

Substituting the boundary conditions R=R 0

whenθ=0 gives:

0 =

1

α

lnR 0 +c

from whichc=−

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 .0 andθ=
50 intoR=R 0 eαθgives the resistance at 50◦C,
i.e.,R 50 = 24 .0e(38×^10

− (^4) ×50)
= 29 .0 ohms.
Now try the following exercise.
Exercise 179 Further problems on equa-
tions 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
   =2 cos^2 y [ tany= 2 x+c]

   
   


3. 
   

   (y^2 +2)

   
   

dy

dx

= 5 y,giveny=1 whenx=

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,

whereL andR 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), wherex is the amount

transferred in timet,kis a constant anda

is the concentration at timet=0 whenx=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 that

Q=Q 0 whent=0.

(b) A circuit possesses a resistance of

250 × 103  and a capacitance of

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

charge falls to 8.0 C. Determine the ini-

tial charge and the charge after 1 second,

each correct to 3 significant figures.

[(a)Q=Q 0 e

−t

CR(b) 9.30 C, 5.81 C]

7. A differential equation relating the difference
   in tensionT, pulley contact angleθand coeffi-

cient of frictionμis

dT

dθ

=μT. Whenθ=0,

T=150 N, andμ= 0 .30 as slipping starts.

Determine the tension at the point of slipping

whenθ=2 radians. Determine also the value

ofθwhenTis 300 N. [273.3 N, 2.31 rads]