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.
dy
dx
= 2 + 3 y
[
x=
1
3
ln( 2 + 3 y)+c
]
dy
dx
=2cos^2 y [tany= 2 x+c]
(y^2 + 2 )
dy
dx
= 5 y,giveny=1whenx=
1
2
[
y^2
2
+2lny= 5 x− 2
]
- 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.
- 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)]
- (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]
- 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]